Aerodynamics

In short

• What it models: the air’s forces on a rocket: the normal force (the sideways push when flying at an angle to the airflow) and the center of pressure (where it acts) from Mach 0 to 5, drag over the same range, and the rolling moment from canted fins and the roll rate. A flight can also take another program’s drag, or its normal force and center of pressure, in place of hpr’s own (The normal force from RASAero II).
• Sources: Barrowman’s 1966 report, 1967 thesis and Centuri TIR-33 (1970), the basis of Barrowman’s method; supersonic linear theory for fins past Mach 1; for drag, mainly Niskanen’s 2009 OpenRocket thesis, with Stoney’s 1961 NASA measurements of noses through Mach 1, and MIL-HDBK-762 (1990) for boattails faster than sound; for tube fins, Weissinger’s ring-wing formula and Fletcher’s 1957 NACA measurements.
• How well it is validated:

  • Faster than sound, drag is only partly validated, and it misses both ways. It reads high against a wind tunnel, most of all for thin, sharp fins; the body alone reads a little low against a handbook’s worked example; and a rocket with a short, steep boattail reads 5% to 15% low against RASAero II. Treat a supersonic flight’s drag, and its apogee, as rough. The bullets below give the numbers.

  • Against OpenRocket, faster than sound, hpr’s supersonic pressure drag (mostly wave drag) is about twice OpenRocket’s, on the one supersonic flight compared, read from OpenRocket’s per-component output but not kept as a record. Which is right is open (#222, a supersonic flight).

  • Base drag under power is unvalidated. hpr takes the burning motor’s area off the base (Niskanen); OpenRocket keeps the whole base. Neither rule has been checked against a measured flight. On one private supersonic flight the choice moves the apogee by about 24 percentage points (#222, Drag).

  • A boattail’s own drag faster than sound against 58 readings of 20 measured boattails of 3° to 10°, Mach 1.2 to 3.12: −21.9% to +28.3%, within 0.0123. Through Mach 1 it reads low, and under Niskanen’s subsonic rule long boattails get almost nothing. Steeper ones in a thick boundary layer read high: +26.4% to +54.2% for 16°. The drag of the base behind a boattail is within 0.0102 of 12 measured bases, which behind a small base can be 40% of its own drag (Boattails faster than sound).

  • Drag reads high against a wind tunnel. Against NASA’s wind-tunnel tests of the Arcas Robin sounding rocket, Mach 0.6 to 4.63, on the forebody only (the models’ bases sat on a sting), every reading is high and 2 of 44 are within 10%, both at Mach 1.0 with the fins on. With the fins on, from Mach 1.5 up, hpr reads +39.4% to +154.0% high: the fins take a blunt edge’s formula. With the fins off it reads +13.5% to +24.1% from Mach 1.5 and +12.0% to +54.1% below, most of it the models’ 15° boattail, which hpr over-predicts in a boundary layer thicker than the boattail is deep. hpr’s base drag behind a plain cylinder has been checked against no measurement faster than Mach 0.3 (Drag against the Arcas Robin wind tunnel).

  • Drag reads low against RASAero II past Mach 1.6, missing M1.8’s 10% there. The curves labelled RASAero in RocketPy’s example rockets don’t record their fins or surface finish, so hpr uses stated guesses for them. Against Calisto’s, the one real RASAero II export, hpr is within 10% at every Mach number up to 0.8, at 3 of the 7 between, and at 8 of 17 from 1.2 to 2.0, where it reads −14.9% to −5.1%, lowest at Mach 2. Other plausible fins bring 14 to 17 of the 17 within 10%, though none puts every row of every band within it; before hpr modeled the boattail’s wave drag it read −29.8% to −24.4% there (Drag against RASAero II through Mach 2).

  • Drag at Mach 0.3 against the same curves: four of seven cases within 10%. Cavour under power (motor burning) is −18.3%, cause open, and Valetudo’s table is 1.44 times its own OpenRocket export.

  • Against MIL-HDBK-762’s worked example, a rocket whose drag the U.S. Army’s handbook calculates term by term with every input known, the fins left out: 6 of 12 Mach numbers within 10%; hpr reads +12.3% to +31.9% from Mach 0.9 to 1.2 (the nose and the base) and −6.0% to −9.6% from Mach 1.6 (friction and the base) (Drag against MIL-HDBK-762’s sample calculation).

  • The normal force and center of pressure at Mach 0 against Barrowman’s worked examples (rockets he calculated by hand), where every center of pressure agrees within 1%, and so does every normal-force slope but his six-fin Recruiter’s: +2.87% high for the rocket and +3.42% for its fins, mostly from a different six-fin rule; and from Mach 0.6 to 4.63 against the same wind tunnel: from Mach 1.5 the slope within +9.4% to −3.3% and the center of pressure within 0.53 calibres (the long model misses the half-calibre target at Mach 1.8 and 2.3, where its body reads high), and between Mach 0.8 and 1.2 both miss (Normal force through Mach 1).

  • The body faster than sound, by the method a flight blends in over 0.3 in Mach from Mach 1.2 at the earliest, on the bodies it covers: against its report’s wind-tunnel measurements of 120 cone- and ogive-cylinders from Mach 3 to 6.28, 117 slopes within ±0.2 per radian and 109 centers of pressure within 0.2 calibres. Against the Arcas Robin wind tunnel’s body alone (its two lengths, the short model and the long), with a pointed nose fitted to its shape and the lip behind its boattail left off (Checking the shock-expansion method):

    • Like for like, fitted at the tunnel’s angles with the body lift a flight adds, the body with its boattail reads +3.4% to +41.0%, within 15% from Mach 3.96 (before M1.8e6 sized body lift and the boattail’s share, 14.9% to 73.2% high). At its 62 angles from 5.5° to 21.7°, 48 are within 15%.
    • The method alone at α → 0 against the tunnel’s line, which includes crossflow: the nose and cylinder of the short model within 5% from Mach 1.8 to 2.96, and −15.0% and −18.7% past Mach 3; the long model −13.7% to −26.4% throughout.

    hpr’s committed Arcas Robin designs fly that method to their base since M1.8e8; like for like, the short model’s body reads 3.02 to 3.95 per radian from Mach 1.5, where the tunnel reads 2.19 to 4.15 (Normal force through Mach 1).

  • In whole flights in wind, body lift, which RocketPy leaves out, is the largest reason a slow rocket’s drift differs from RocketPy’s (ADR-026). Nothing against a real flight.

  • Roll: the spin that canted fins give, against NASA’s measured roll effectiveness (the rolling moment per degree of cant), is within 5.3% at all 8 readings from Mach 2.3 to 4.63, and 14.3% to 47.8% high at Mach 1.5 and 1.8; the roll damping reads 5.9% to 16.2% low against the one measured set, that of the Basic Finner, a standard finned test body, from Mach 1.5 to 3 (Roll against the Arcas Robin and the Basic Finner).

  • Another program’s normal force, read from RASAero II’s export: every row of Calisto’s export comes back from hpr’s table, and a flight on a table swings in pitch as the equations predict; no real export has flown faster than Mach 0.75 (The normal force from RASAero II).

  • Pods take each pod’s parts on Barrowman’s rules, once per pod. They are checked against hand-worked numbers, and against OpenRocket on six probe designs, within 0.81% of its apogee below Mach 0.81. No measured flight checks them. Both codes leave out the pods’ and the body’s effect on each other’s flow, and hpr a single pod’s off-axis moments (Pods).

  • Tube fins, each tube a ring wing below Mach 0.8: the slope is within 3% of five rings measured in a wind tunnel. No tube fin rocket has been checked against a measurement. On OpenRocket’s example, hpr’s apogee is 6.95% higher, a net gap in the drag, and OpenRocket’s tube-fin drag was refined against real flights, so hpr’s probably reads low. The center of pressure rests on a judgement that moves that example’s margin from 0.29 to 0.79 calibres, against OpenRocket’s 1.87. The gap to OpenRocket is measured on 14 probe designs and pinned by a test; nothing measured says which code is nearer (Tube fins, #228).

• What it leaves out: large angles and stall, though a flight uses these models at every angle. Faster than sound (transonic and supersonic), a steep boattail’s drag in a thick boundary layer reads high, and nothing corrects for it; the fins’ drag takes a blunt edge’s formula, which reads far high for thin, sharp fins, and nothing models a thin fin’s own wave drag or the drag where fins meet the body. Faster than sound a flight takes a pointed nose and its cylinder from the method that adds the cylinder’s lift, and a boattail behind them from a measured correlation of boattails of 4° to 9.5°, an extrapolation for steeper ones, which hpr stops reading past the angle where the flow separates (The body faster than sound in a flight). A nose with a vertical tip (power-series, Haack, elliptical) takes a Newtonian cap ahead of the method, checked on a sphere-cone only (Blunt tips), and a lip inside a boattail’s wake carries nothing (A lip in a boattail’s wake). A conical flare flush with the tube ahead of it flies the method too, and ends the run, while its corner’s shock stays attached, checked against one measured flare, where it reads −1.9% and +7.0% at Mach 1.9 and 2.3, +13.4% at 2.96, then +51.5% and +50.4% at 3.95 and 4.63 (What a marched flare is worth); a flare shallow enough to turn the flow almost not at all is read by an older, rougher method instead, which nothing measures (A near-flat flare). A rocket with any other widening shape, or any step, behind the nose (a motor retainer behind a step down counts: the step ends the run) keeps slender-body theory for its whole body at every speed, which reads low past Mach 3 (A step in radius: −8.65% and 1.03 calibres at the threshold). Body lift leaves out the fall in crossflow drag past the critical crossflow Reynolds number (Body lift), and it reads too large at the few degrees a slope is fitted over: the body alone misses the 15% target the milestone set on six of eleven wind-tunnel rows, by +37.7% at worst and within 5% at Mach 3.96 and 4.63 (The body alone, against the 15% target). There are no damping coefficients for pitch and yaw: a flight takes that damping from each part’s own local flow. The roll forcing near Mach 1.5 reads high, and nothing measured checks roll below it (Roll: forcing and damping).

Code and sources

Code: hpr_aero::body (bodies of revolution), hpr_aero::crossflow (body lift), hpr_aero::fins (fin sets), hpr_aero::nose_drag (noses’ drag through Mach 1), hpr_aero::afterbody (boattails faster than sound), hpr_aero::shock_expansion (the body faster than sound), hpr_aero::blunt_tip (a blunt tip’s cap and its handover), hpr_aero::table (tables from other programs), hpr_aero::tube_fins (tube fins) and hpr_aero::model (a whole rocket’s terms, built from its Layout). Decisions: ADR-008 (normal force and center of pressure) and ADR-009 (drag). The milestone M1.5a covers the subsonic normal force and center of pressure, M1.5b the subsonic drag and override tables; M1.8a the normal force through Mach 1 (ADR-027); M1.8b1 the drag through Mach 1 (ADR-028); M1.8b3 boattails faster than sound (ADR-030); M1.8c roll (ADR-031); M1.8d the normal force from RASAero II (ADR-032); M1.8e1 the body faster than sound (hpr_aero::shock_expansion, ADR-033), flown from M1.8e2, M1.8e4 the boattail’s share of it, and M1.8e6 body lift’s size at every speed and the boattail’s measured share (hpr_aero::supersonic_boattail, ADR-037). The rest of transonic and supersonic flow arrives with the rest of M1.8, the supersonic aerodynamics milestone.

A Loft lesson is something learned from Loft, the project that came before hpr-sim: usually a mistake it made, sometimes a check worth keeping. This page names the ones that concern aerodynamics, and the test here that covers each.

Sources:

• [B66] J. S. and J. A. Barrowman, The Theoretical Prediction of the Center of Pressure, NARAM-8, 1966 (barrowman-1966-naram8-nakka; the Apogee copy lacks pp. 39–50).
• [B67] J. S. Barrowman, The Practical Calculation of the Aerodynamic Characteristics of Slender Finned Vehicles, MS thesis, 1967 (NASA/TM-2001-209983).
• [TIR] J. S. Barrowman, Calculating the Center of Pressure of a Model Rocket, Centuri TIR-33, 1970.
• [N09] S. Niskanen, Development of an Open Source model rocket simulation software, MSc thesis, 2009, chapter 3.
• [TD] OpenRocket technical documentation 13.05 (the thesis revised; a document, not code).
• [G] R. Galejs, Wind Instability: What Barrowman Left Out, Sentinel 39.
• [762] MIL-HDBK-762(MI), Design of Aerodynamically Stabilized Free Rockets, 1990.
• [TN2114] S. M. Harmon and I. Jeffreys, Theoretical Lift and Damping in Roll of Thin Wings with Arbitrary Sweep and Taper at Supersonic Speeds: Supersonic Leading and Trailing Edges, NACA TN 2114, 1950.
• [D4013] J. C. Ferris, Static Stability Investigation of a Single-Stage Sounding Rocket at Mach Numbers from 0.60 to 1.20, NASA TN D-4013, 1967.
• [D4014] C. D. Babb and D. E. Fuller, Static Stability Investigation of a Sounding-Rocket Vehicle at Mach Numbers from 1.50 to 4.63, NASA TN D-4014, 1967.
• [S61] W. E. Stoney, Collection of Zero-Lift Drag Data on Bodies of Revolution from Free-Flight Investigations, NASA TR R-100, 1961 (nasa-tr-r-100-stoney-1961).
• [J53] J. R. Jack, Theoretical Pressure Distributions and Wave Drags for Conical Boattails, NACA TN 2972, 1953.
• [CS51] E. M. Cortright Jr. and A. H. Schroeder, Investigation at Mach Number 1.91 of Side and Base Pressure Distributions over Conical Boattails without and with Jet Flow Issuing from Base, NACA RM E51F26, 1951.
• [DN54] C. A. de Moraes and A. M. Nowitzky, Experimental Effects of Propulsive Jets and Afterbody Configurations on the Zero-Lift Drag of Bodies of Revolution at a Mach Number of 1.59, NACA RM L54C16, 1954.
• [MJ54] B. Moskowitz and J. R. Jack, Aerodynamics of Slender Bodies at Mach Number of 3.12 … V: Aerodynamic Load Distributions for a Series of Four Boattailed Bodies, NACA RM E54B11, 1954.
• [C57] J. M. Cubbage Jr., Jet Effects on the Drag of Conical Afterbodies for Mach Numbers of 0.6 to 1.28, NACA RM L57B21, 1957.
• [Love57] E. S. Love, Base Pressure at Supersonic Speeds on Two-Dimensional Airfoils and on Bodies of Revolution with and without Fins Having Turbulent Boundary Layers, NACA TN 3819, 1957.
• [C72] W. B. Compton III, Jet Effects on the Drag of Conical Afterbodies at Supersonic Speeds, NASA TN D-6789, 1972.
• [R1135] Ames Research Staff, Equations, Tables, and Charts for Compressible Flow, NACA Report 1135, 1953.
• [SD56] C. A. Syvertson and D. H. Dennis, A Second-Order Shock-Expansion Method Applicable to Bodies of Revolution Near Zero Lift, NACA TN 3527, 1956 (also NACA Report 1328; naca-tn-3527-syvertson-dennis-1956).
• [S64] J. L. Sims, Tables for Supersonic Flow Around Right Circular Cones at Small Angle of Attack, NASA SP-3007, 1964 (nasa-sp-3007-sims-1964-cones-small-alpha).
• [J77] L. H. Jorgensen, Prediction of Static Aerodynamic Characteristics for Slender Bodies Alone and With Lifting Surfaces to Very High Angles of Attack, NASA TR R-474, 1977 (nasa-tr-r-474-jorgensen-1977).
• [WP68] W. D. Washington and W. Pettis Jr., Boattail Effects on Static Stability at Small Angles of Attack, U.S. Army Missile Command report RD-TM-68-5, 1968 (washington-pettis-1968-rd-tm-68-5).
• [J68] C. M. Jackson Jr., W. C. Sawyer and R. S. Smith, A Method for Determining Surface Pressures on Blunt Bodies of Revolution at Small Angles of Attack in Supersonic Flow, NASA TN D-4865, 1968 (nasa-tn-d-4865-jackson-1968).
• [S62] A. Seiff, Secondary Flow Fields Embedded in Hypersonic Shock Layers, NASA TN D-1304, 1962 (nasa-tn-d-1304-seiff-1962).
• [R22] C. E. Rogers, RASAero II Comparisons with ARCAS Center of Pressure (CP) and Drag Coefficient (CD) Wind Tunnel Data, Rogers Aeroscience, 2022 (slides).
• [RAS] C. E. Rogers and D. Cooper, Rogers Aeroscience RASAero II Aerodynamic Analysis and Flight Simulation Program Users Manual, version 1.0.2.0, 2019.
• [F57] H. S. Fletcher, Experimental Investigation of Lift, Drag, and Pitching Moment of Five Annular Airfoils, NACA TN 4117, 1957 (naca-tn-4117-fletcher-1957-annular-airfoils).
• [H65] S. F. Hoerner, Fluid-Dynamic Drag, 1965, p. 7-13, Ring Foil, and p. 3-12, Fig. 20, the forebody pressure drag of blunt heads (hoerner-1965-fluid-dynamic-drag).
• [HB85] S. F. Hoerner and H. V. Borst, Fluid-Dynamic Lift, 1985, p. 19-16, Ducted Body (hoerner-borst-1985-fluid-dynamic-lift).
• [W21] N. Wagner, Theoretical and Experimental Investigation into the Flight of an X-Zylo, arXiv:2102.02647, 2021, eq. 15, quoting J. Weissinger, Zur Aerodynamik des Ringflügels, 1955 (arxiv-2102.02647-x-zylo-ring-wing).
• [PNK57] W. C. Pitts, J. N. Nielsen and G. E. Kaattari, Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds, NACA Report 1307, 1957 (naca-report-1307-pitts-nielsen-kaattari-1957-wing-body-tail-lift).

Conventions

An aerodynamic coefficient is a force divided by the dynamic pressure q (the pressure of the oncoming air, ½ ρ V², with ρ the air’s density and V the airspeed) and by the reference area. It has no units, so the same number describes a small rocket and a large one of the same shape. These are the symbols the whole page uses; each section defines its own as well.

The rocket’s CP is the average of its components’ CPs X_i, each weighted by that component’s slope: X = Σ C_Nα,i X_i / Σ C_Nα,i ([B66] p. 38; [N09] eq. 3.29). The components i are the nose cones, transitions, body tubes and fin sets; a body tube’s own slope is 0 (Bodies of revolution, below). A rocket whose slopes add up to zero has no CP.

In code, these are the names the page uses:

Your rocket’s center of pressure

hpr works out the CP from the rocket’s shape alone, the way Barrowman’s method does by hand. Each nose, transition and fin set gets its own slope and CP (the next two sections), and the rocket’s CP is their weighted average, as above. Your own rocket prints the CP, the center of gravity and the stability margin of an example rocket.

To get it in code:

1. Take the rocket’s AeroModel from a simulation with Simulation::aero, or build one from a design with Rocket::layout and AeroModel::new.
2. Call AeroModel::normal_force with Flow::axial(mach): the air straight along the axis, at a Mach number below 5. The result’s cp_station_m is the CP, in meters aft of the nose tip, and its slope_per_rad is the rocket’s C_Nα.
3. AeroModel::components, at the same flow, lists each component’s share, which shows what moves the CP.

What changes it:

• Speed. Below Mach 1.2 only the fins’ terms change with Mach number; past it the bodies’ can too (The body faster than sound in a flight). Up to Mach 0.8 the fins’ slope grows through the Prandtl–Glauert factor, the classic correction for the air’s compressibility, whose effect grows as the speed nears that of sound (Prandtl–Glauert, under Fins). How much a fin set gains depends on its span, area and sweep. So as the rocket speeds up, the CP moves toward its fins:

  • With fins only at the tail, it moves aft.
  • With canards (a second fin set near the nose) as well, both sets gain, and the CP can move either way, depending on each set’s shape and place.
  • Up to Mach 0.8 hpr keeps each fin set’s own CP a quarter of the way along its mean aerodynamic chord (MAC, a weighted average of its chords). From there it moves aft. The fins’ slope grows up to where supersonic theory starts, Mach 1.2 or later, and falls past it (Fins through Mach 1), so a fast rocket’s CP moves forward again.

  Flow::axial(0.0) gives the low-speed CP that Barrowman’s method gives by hand.

• Angle. Flow::axial gives the small-angle CP. At an angle of attack, body lift adds a force at each body’s side-view centroid, the center of its outline seen from the side (Bodies of revolution, below), and the CP moves with it.

• Stability. A rocket is statically stable when its CP is aft of its center of gravity (CG).

  • Assembly::mass_properties gives the rocket’s mass, CG and inertia t seconds after ignition. A simulation’s assembly comes from Simulation::assembly.
  • Its cg_m is the CG in body axes, so the CG’s station is −cg_m.z.
  • The CP’s station less the CG’s, divided by d_ref, is the stability margin in calibres.
  • Flight metrics gives the margin from the rail exit to apogee or the first deployment, with the air along the axis, at Mach 0 (the static margin) and at the flight’s Mach number (the flight margin), and gives none where the slopes all but cancel.

Bodies of revolution

Nose cones, transitions and body tubes, from the outer profile. A shoulder (the sleeve of a nose or transition that slides into the next tube) is inside the body and adds nothing. For one component:

Body lift is the extra push of the air crossing the body at larger angles of attack. It grows with sin² α, so it is zero at α = 0 and adds nothing to the slope there. Its size is in Body lift, below.

• A nose with a sharp tip has slope 2. A cylinder has 0 and no CP. A boattail has a negative slope, and the CP formula for a frustum (a cone with its tip cut off), [B66] eq. 44, still holds ([B66] p. 21).
• Radius steps (an extrapolation). A step where one body component meets the next adds (2/A_ref)ΔA at the joint, the limit of a transition whose length goes to zero, so the body’s total slope is [B66] eq. 10 over the whole body. [B67] p. 18 assumes no discontinuities, so this goes beyond the source; leaving the step out would silently drop its slope (a 27 mm nose base on a 29 mm tube loses 13%). It is reported with the aft component (BodyAero::step_area_m2). A blunt front face gets no term, as eq. 10 gives. The design checks warn about steps (radius_step); the real flow separates there, and the drag buildup counts it as a zero-length shoulder or boattail (Steps in radius, under Drag).
• V and the planform come from integrating the real profile (hpr_design::revolve), so ogive, power, parabolic and Haack transitions (Shapes) get their own CP (Loft lesson L9: Loft used the conical transition’s CP formula for every shape). [B66] puts a tangent ogive nose’s CP at 0.466 L instead, with L the nose’s length: 0.2–0.9% different at fineness 2.8–5.
• Slender-body theory’s slope has no Mach term: [B67] p. 18 leaves body compressibility out as a conservative choice, and [N09] p. 22 takes the body’s normal force as the same at all speeds. Faster than Mach 1.2 a flight can use another method (The body faster than sound in a flight).
• Body lift is zero at α = 0, so Barrowman’s worked examples don’t test it.

Body lift

Changed in M1.8e6 (ADR-037): until then hpr used Galejs’s constant, C_N = K (A_plan/A_ref) sin² α with K = 1.1 at every speed.

What this covers: the size of body lift, the sideways push of the air crossing a body at an angle of attack. How far to trust it: it is Jorgensen’s method ([J77]) with hpr’s own way of combining two of his figures (below). Against NASA’s Arcas Robin body alone at the 62 angles the wind tunnel plotted from 5.5° to 21.7°, from Mach 1.5 to 4.63, hpr’s normal force with it is within 15% at 48 (34 with Galejs’s constant); where the air crosses the body faster than sound, it reads 1% to 16% high. Below Mach 1 the only check is that body’s slope fitted from −4° to +4°, where body lift adds a little: at Mach 0.6 hpr’s body reads 25% high (41% with Galejs’s constant), against readings the tunnel determines poorly. No real flight checks it: the real flights so far compare heights, not drift, so whether it is better than Galejs’s constant for a slow rocket leaving the rail in wind, where it matters most, is open.

At an angle of attack α the air meets the body partly from the side, at V sin α. Behind a long cylinder in a cross-wind the air separates, and the drag of that separated flow pushes the body sideways. Jorgensen sizes body lift from that drag: C_dn, the drag coefficient of an infinitely long circular cylinder in the crossflow, and η, the ratio of a finite cylinder’s to an infinite one’s. Both depend on the crossflow Mach number M_n = M sin α, the Mach number of the air crossing the body; η also on the body’s fineness f, its length over its largest diameter.

• C_dn is Jorgensen’s value below the critical crossflow Reynolds number, where the air separates from a smooth cylinder early: “C_dn = 1.2” at low speed (p. 15). From M_n 0.6 to 1.2 hpr takes the points Jorgensen marks as extrapolated from NASA Ames wind-tunnel data, and from 1.4 his curve through the supersonic experiments.
• η. (Jorgensen’s; the shock-expansion method below uses η for something else.) Fig. 4 gives η against length for cylinders measured only at low speed. Fig. 6 gives how η grows toward 1 as the crossflow speeds up, but only for the two bodies (fineness 10 and 12) it was computed from: Jorgensen divided the η C_dn those bodies’ measured normal force gives (his Fig. 5) by Fig. 1’s C_dn. For any other fineness hpr scales Fig. 6’s η by how much Fig. 4 changes it for the body’s length, and lets that scaling fade by the share s Fig. 6 has risen toward 1. The share it uses, r, never falls back: Fig. 6 dips at M_n = 1 only because Jorgensen divided by Fig. 1’s peak there, not because length counts again. Below M_n 0.8, where Fig. 6 only rises, this is η = η₄ + (1 − η₄) s; past 0.8 every fineness takes Fig. 5’s η C_dn to within 0.3%. That rule is hpr’s; it gives Fig. 6 back for a body of fineness about 10.6 and stays below 1.
• Sampled, not smoothed. Near M_n = 1 Fig. 1’s C_dn peaks and Fig. 6’s η dips, each steeply. hpr reads both at the eleven crossflow Mach numbers Jorgensen computed Fig. 6 at and interpolates between them, so their product is his own η C_dn there: within 3% of his Fig. 5 at all ten of its points from 0.5 to 1.6 (test the_product_follows_figure_5).

A worked example. NASA’s short Arcas Robin model without fins, fineness 18.18, has a planform of 20.79 times its cross-section. At Mach 2.3 and α = 12.56°, M_n = 2.3 × sin 12.56° = 0.50. Fig. 4 gives η₄ = 0.742; Fig. 6 gives η₆ = 0.804, so s = (0.804 − 0.69)/0.31 = 0.368 and η = 0.742 + 0.258 × 0.368 = 0.837. With C_dn = 1.334, η C_dn = 1.117, and body lift is 1.117 × 20.79 × sin² 12.56° = 1.10. Adding the attached-flow part (the method’s slope at small angles, without crossflow), hpr’s body gives C_N 1.632 there; the tunnel measured 1.630. This point happens to agree closely; across the 62 points the spread is −22.3% to +31.7% (table below).

What changes in a flight. At the low crossflow speeds of most flights η C_dn is 1.2 η₄(f): 0.82 at fineness 10, 0.90 at 20, 0.95 at 30, against Galejs’s 1.1. A slow rocket leaving the rail in wind feels it most (Validity and open questions). The tunnel points where hpr reads high, the air crossing the body faster than sound, are at 12° to 21° from Mach 2.3 up; a flight meets such angles that fast only if it is unstable or hit by a strong gust.

How it was checked. The Arcas Robin wind tunnel measured the body alone from −5° to 21° at six Mach numbers (TN D-4014; the points above +4° were read for this milestone into arcas-robin-high-alpha.json). arcas-robin-crossflow.json compares hpr’s body at each point, with the pointed nose fitted to the tunnel’s and the lip left off (Checking the shock-expansion method):

At the lowest crossflow speeds the readings can’t say how much of what is left is body lift and how much the slope at α → 0 (Checking the shock-expansion method). Where the air crosses faster than sound, hpr’s η C_dn (1.45 to 1.61) is above what the tunnel’s points need (1.26 to 1.54). Jorgensen’s η C_dn was worked out from measured normal force less his own attached-flow term, sin 2α cos(α/2); hpr pairs it with its own, sin α times its slope, which is 8% larger at 20° and, faster than sound, carries the method’s slope for the nose and cylinder, 2.55 to 3.40 against slender-body theory’s 2. Two cautions from Jorgensen: his η comes from cylinders measured “only at very low subsonic Mach numbers” (p. 17), and the tunnel tripped its boundary layer, which can move the flow past the critical crossflow Reynolds number, where C_dn falls to “between about 0.15 and 0.30” (p. 15).

What it leaves out. That fall past the critical crossflow Reynolds number (about 2 × 10⁵, [J77] Fig. 2), which Jorgensen computes only for illustration; roughness and fins’ effect on the body’s crossflow; and Jorgensen’s own attached-flow term, sin 2α cos(α/2) in place of hpr’s sin α. Galejs’s constant stays available for comparison (BodyLift::Galejs, with AeroModel::with_body_model): [G] cites Hoerner’s 1.1 to 1.5 and fitted 1.0 to his own data.

Bodies faster than sound

Flown faster than sound for a pointed nose and its cylinder (see The body faster than sound in a flight, below). This is M1.8e1’s method, the second-order shock-expansion method as a tested library model. Against its report’s wind-tunnel data, 117 of 120 slopes are within 0.2 per radian. On the Arcas Robin’s body it reads from 16% high at Mach 1.5 to 27% low past Mach 3 at α → 0, a comparison that leaves out the body lift a flight adds; compared the way the tunnel measures, the body reads high (Checking the shock-expansion method, under Verification). It needs a pointed tip; a blunt or vertical one (power-series, elliptical and Haack noses) takes a Newtonian cap ahead of it (Blunt tips, below).

Slender-body theory, above, gives a pointed nose a slope of 2 and a cylinder none, at any speed. Faster than sound that is too little. The air speeds up around the shoulder where the nose meets the cylinder, and its pressure then recovers along the cylinder toward the free stream’s. At an angle of attack it recovers unevenly around the body, so the cylinder carries lift too, more the longer it is. NASA measured the Arcas Robin’s body alone at 3.9 per radian at Mach 3.96, where slender-body theory gives 2 for its nose.

hpr computes that lift by Syvertson and Dennis’s second-order shock-expansion method ([SD56]), for a body with a pointed tip and supersonic flow everywhere on it, as the slope at small angles of attack (α → 0). The older generalized shock-expansion method holds the pressure constant along each straight piece of the profile; the second-order method also carries the pressure’s rate of change across each corner, so the pressure can recover along a piece:

1. The tangent body. The profile becomes straight elements, each tangent to it: the first at the tip, the rest at equal steps along a curved nose (ten per curved piece, the report’s own choice), one per cone or cylinder. Where two elements meet is a corner.
2. The tip is a cone, so its flow is exactly a cone’s, found by integrating the Taylor–Maccoll equation (the exact equation of supersonic flow over a cone) from the shock to the surface ([R1135] eq. 177). The shock must be attached, touching the tip, which holds up to a half-angle that grows with the Mach number. For cones under 0.03°, where that equation can’t be integrated, hpr takes slender-cone linear theory, blended in up to 0.06°.
3. Around each corner the flow turns by a Prandtl–Meyer expansion.
4. Along each element the pressure relaxes from its value behind the corner toward the pressure on the element’s tangent cone: the cone, pointed into the oncoming flow, whose surface has the body’s local slope there. The lift per unit length relaxes the same way, toward that cone’s.
5. The slope and CP follow by adding the lift over the body.

Here δ is an element’s angle to the axis, p the pressure over the free stream’s (the undisturbed air ahead of the rocket), M the Mach number at the surface, r the radius at the corner, Ω how much a thin tube of flowing air widens as its Mach number rises (its area over its area at Mach 1, [SD56] eq. 7), μ the Mach angle asin(1/M), and (dC_N/dα)_tc the slope of the tangent cone, digitised by hand from the report’s Fig. 2 into a table and continued past that chart’s 24° by Sims’s own tables of the same theory, to 30° (ADR-042, cone_normal_force_slope). For a worked example with numbers, see Checking the shock-expansion method.

• A cylinder’s tangent cone is the free stream, so its lift decays to nothing along it.
• A boattail has no tangent cone. The report’s footnote 8 takes the free stream’s pressure and a slope of 2, “reasonable results for bodies having moderate amounts of boattail”. The method does the same, but a measurement says it takes too little lift off: since M1.8e6 a flight gives a boattail a measured share instead (The body faster than sound in a flight).
• Where the method stops. The relaxation holds only where the gradient behind a corner points toward the tangent cone’s pressure (η ≥ 0, [SD56] p. 13). The report states that as a condition and doesn’t say how it went on where it fails, near a sharp tip at high Mach number. hpr’s own reading is to reduce such an element to the older generalized method, which the report says the equations become at η = 0: the pressure stays as it is along the element and no gradient passes to the next corner. On the report’s fineness-3 ogive at Mach 5.05 that departs from its values (issue #81: the method’s limit near a sharp tip). hpr reads an element that way wherever it has a tangent cone of its own, behind the nose as well as on it (A near-flat flare). A cylinder’s tangent cone is the free stream and a boattail’s is footnote 8’s, so neither is a solution of that element’s own flow: one of those that would need reducing is refused instead, and the whole body keeps slender-body theory (issue #123: a cylinder’s or a boattail’s reduced element).
• Mach number over nose fineness. The report states the method for 0.4 to 2; hpr doesn’t enforce it (the report’s own Mach 6.28 rows are at 2.09, and the Arcas Robin at Mach 1.5 is at 0.36).
• Its range. The report states the method for Mach number over nose fineness from 0.4 to 2, within ±0.2 per radian and ±0.2 calibres of its measurements. Fig. 2 covers Mach 3 to 10; below Mach 3 hpr holds the Mach 3 curve, an assumption. The tip’s shock must be attached, and the profile continuous. Tangent cones run to 30°: to 24° from Fig. 2, and from there to 30° from NASA SP-3007’s tables of the same theory, which agree with the chart to 0.0021 per radian where both cover the same angle (ADR-042). A cone steeper than 30° is refused, and the whole body then keeps slender-body theory.
• What it leaves out: the crossflow lift that grows with sin² α (Body lift, above), and anything viscous. It is the slope at small angles only.

The body faster than sound in a flight

What this covers: how a flight uses the method above, from Mach 1.2, and a boattail’s measured share. How far to trust it: as far as the method’s own checks above, for a pointed nose and cylinder. A boattail behind them takes Washington and Pettis’s measured increment ([WP68]), which their data give within about 15% for conical boattails of 4° to 9.5°; a steeper, shorter, longer or narrower one, like the Arcas Robin’s 15°, is an extrapolation, as is a transition that isn’t conical (it takes the same correlation from its length and radii). A long boattail reads the curve near zero argument, which comes from the report’s lowest supersonic runs. Past 16°, where the flow separates, hpr stops reading the correlation any steeper and holds it there (A steep boattail reads the correlation no steeper than 16°, issue #90: how steep a boattail the correlation should cover); nothing measures what such a boattail really carries, and the choice is worth 0.67 to 1.35 calibres of center of pressure at 30°, most at the lowest speeds. A tube behind the boattail takes the method’s decay of its expansion, which no measurement here checks. A blunt or vertical nose tip takes a Newtonian cap ahead of the method (Blunt tips), an extrapolation from spherical caps, and a lip inside a boattail’s wake rides along carrying nothing (A lip in a boattail’s wake). A conical flare flush with the part ahead of it flies the method while its corner’s shock is attached, and is read drawn out where it is not; one measured flare has been put beside it (What a marched flare is worth), and the drawn-out half still has none. A rocket with any other widening shape, or any step, behind the nose gets nothing from the method and keeps slender-body theory, which on the Arcas Robin’s body reads 15% to 50% below the tunnel faster than sound. The join between the two models is a judgement, not a measurement, and no validation flight goes past Mach 1.06 (Prometheus, the fastest; see its max_mach rows in the validation report), so no flight checks it yet.

Which model your rocket gets. Every body takes body lift at every speed. Past Mach 1.2, a rocket whose first body is a nose (pointed, or with a blunt or vertical tip that the cap covers), followed only by tubes of its radius and boattails (and tubes behind those), takes the method below for those parts, and each boattail its measured share. A lip wholly inside a boattail’s wake rides along, carrying nothing; one only partly in the wake gets the method in the wake’s own proportion, and slender-body theory for the rest (A lip in a boattail’s wake). A conical flare flush with the part ahead of it joins the run as well, and ends it (A flare through the method); a widening part behind a boattail is a lip, not a flare, and keeps the lip’s rule. Anything else (a step, a widening shape that is not a cone, a motor retainer behind a step down (the step itself ends the run) or a nose steeper than the cap’s handover all the way to its base) keeps slender-body theory for its whole body at every speed.

Where that choice still jumps. It is one choice for the whole body, so wherever it turns on a threshold, a rocket either side of that threshold gets two different models, and the difference is the whole body’s, not the part that changed. At Mach 3 and 4°, each remaining threshold is worth this much. The first four rows are measured on two test bodies: the flare behind a boattail on a finless body, so its share is of a body’s normal force alone, and the other three on the tests’ straight rocket, which keeps its four fins, so the two kinds of share don’t compare. The last column says who owns it: open means an issue with no milestone behind it, queued a milestone on the roadmap, and no longer a switch a threshold since removed, kept here because its size is the measured cost of the model it replaced (issue_87s_switches_are_this_big pins the first four, a_lip_in_a_boattails_wake_carries_nothing the lip’s two, and a_near_flat_flare_marches_every_row_and_the_fallback_is_still_measured the last):

In the first four rows the center of pressure moves aft when the method is lost, so a rocket that trips one reads more stable than one that doesn’t. The two lip rows and the near-flat-flare row go the other way: on a body whose tail takes lift off (a boattail’s wake, or a flare the method reads lower than slender-body theory does), slender-body theory puts the center of pressure forward of the method’s, so losing the method there reads less stable. Which way it goes depends on the body; what is reliable is the size.

The lip row about length is a switch the wake’s grading does not cover: a lip is only sheltered if it is no longer than the boattail’s drop in diameter, which is the wake’s own scale, and that length is a threshold, not a ramp. It is the one this page’s measurements use to take a rocket off the method without changing a radius or an angle.

The first lip row is no longer a switch: it is spread over the band the wake grades, as above. The second still is one. Nor is the last: since M1.8e19 the march reads that flare’s element rather than refusing it, and what is left where it used to switch is the loading’s step at the corner’s crossing: +0.129% and 0.0051 calibres on this rocket, but larger on other bodies and on longer flares, in A near-flat flare. The two tips moved rather than went: a pointed tip’s edge is the cone tables’ 30° since M1.8e11, and a vertical tip’s is the handover’s cap, which stands at 24° for the reason in What the cap is worth.

The flare went (M1.8e17, ADR-047): a conical flare in the free stream now flies the method, and the boundary where its shock detaches has nothing jumping across it; see A flare through the method below. The second row is what is left of it, a flare behind a boattail, which is a lip in the boattail’s wake rather than a flare in the free stream and keeps its own rule. The last row is what that milestone turned up on the way, and M1.8e19 closed it: a band of near-flat flares, a third of a millimeter tall, whose one element the march reduces to the generalized method.

The step stayed, and M1.8e15 says why: the march needs a profile without a jump in it, so a step needs a model of its own rather than a decision about an existing one, and the obvious fix turned out to cost more than it saved. What that milestone did measure is in A step in radius below. Its threshold is finer than it sounds: a billionth of the radius, a few hundredths of a nanometer on a 54 mm body, so any step a person would draw is past it, and a rocket whose shape sits near one of these thresholds is worth checking on both sides.

What a flight takes. The method covers the nose, when it is the first body (a blunt or vertical tip behind its Newtonian cap), the body tubes straight behind it at the same radius, and boattails (transitions that narrow toward the tail) and tubes behind those, with no step between them (M1.8e4), and a conical flare, which ends the run (M1.8e17). It flies only if nothing else behind them changes the radius: no step, and no widening shape but that flare. Mixing the method’s nose and cylinder with slender-body theory’s boattail would put the center of pressure further off than slender-body theory alone, so the boattail takes a supersonic share too, measured (below). Each covered part gets its own share, so the flight’s pitch damping still comes from each part’s own local flow. That flow is taken at one station per part; the part’s force and its moment about the nose tip are the method’s either way. A nose or cylinder takes that station at its share’s own center of pressure. A boattail’s share is usually negative, and so is a tube’s behind it: the method carries the boattail’s expansion down the tube, where it fades out over several calibers, so a long tube can lose as much as the short boattail. On the finned rocket of the tests (measured by hand, not pinned), a 5.7° boattail 0.05 m long takes 0.284 per radian off at Mach 2 (footnote 8 took 0.117) and the 0.3 m tube behind it 0.210. Such a share could cross zero as the Mach number changes, and its center of pressure would then run off to infinity. So these parts take their local flow where slender-body theory does, on the part: a boattail at its slender-body center of pressure, a tube at its body-lift station. Only the damping feels this: on the test rocket at Mach 2 the tube’s share acts at 1.040 m but its flow is taken at about 1.15 m, so with the center of mass near 0.7 m that part’s (negative) damping reads about 30% large. Body lift, the sin² α term, is unchanged.

The boattail, measured. Washington and Pettis ([WP68]) mounted the aft section of a wind-tunnel model on its own balance and measured it with a conical boattail and as a plain cylinder of the same length, Mach 1.75 to 4.5, and a whole model with and without one from Mach 0.8 to 1.5. The boattail’s increment in slope collapses onto one curve:

ΔC_Nα / [1 − (D_B/D)²] = F(√(M² − 1) / (L_B/D)), per degree on the cylinder’s area,

with D the diameter ahead of the boattail, D_B its base diameter and L_B its length ([WP68] Fig. 5, p. 8, read by hand into WP_SLOPE_PER_DEG to about ±0.02 per radian). The increment acts about halfway along the boattail, from 43% of its length at Mach 2 to 64% at 4.5 ([WP68] Fig. 6, p. 9). A flight gives a boattail the share the method gives a cylinder of the boattail’s length and fore diameter in its place, plus that increment at that center of pressure (test the_boattail_takes_washington_and_pettis_increment). Slender-body theory gives the same boattail 2[(D_B/D)² − 1] at every speed, which is Fig. 5’s own subsonic line; faster than sound the measured increment is less than half of it at the Arcas Robin’s speeds (0.24 to 0.47), and footnote 8’s less again, though the curve is not always below that line: across Fig. 5 it runs from 0.23 to 1.58 times it, passing it at an argument of 0.635, near Mach 1.

A steep boattail reads the correlation no steeper than 16°. Washington and Pettis measured boattails of 4° to 9.5°, where the flow follows the surface. Past about 16° it doesn’t: the drag buildup already treats a boattail as separating from there (Boattails faster than sound, Cubbage’s steepest attached boattail). Nothing measures what a separated boattail’s normal force then does, so hpr reads the correlation at the steepest angle where the flow is still attached: a boattail past 16° takes the increment of one of the same radii drawn out to 16°, and its center of pressure stays on the real boattail (ADR-040, issue #90: how steep a boattail the correlation should cover). The Arcas Robin’s 15° boattail is untouched; Calisto’s 18.4° reads its correlation as a 16° one. The increment is continuous in the angle, so a rocket doesn’t jump as its boattail is drawn steeper (test a_separating_boattail_reads_the_correlation_at_its_steepest_measured_angle).

Why hold it rather than let it fade. The two honest limits for a separated boattail are the correlation held at 16°, and nothing at all, the body behaving as though the boattail were a cylinder, since a separated surface no longer turns the flow. hpr takes the first. The increment is negative, so it takes lift off the tail: holding it keeps the center of pressure forward, and letting it fade to zero would move the center of pressure aft and make a steep boattail look more stable than anything measured. On the tests’ rocket (an ogive nose, a tube, and a 30° boattail), the body’s center of pressure sits this much further aft if the increment fades away than if it is held:

That is the size of the doubt, and it is largest where a hobby rocket spends its supersonic flight: a boattail steeper than 16° is worth two thirds of a calibre at Mach 4.63 and a third of a calibre more than one at Mach 1.5. hpr takes the forward end of that range (test a_separating_boattail_reads_the_correlation_at_its_steepest_measured_angle pins both ends).

There is one more bound, on the holding rather than on the measurement. Reading a longer boattail walks the correlation’s argument √(M² − 1)/(L_B/D) toward zero, where Fig. 5’s curve comes from the report’s lowest supersonic runs and rises past Munk’s slender-body line, which the report plots there for comparison at subsonic speeds (p. 3). hpr does not invent a length and then read that branch, so the extra the holding takes off stops at potential flow’s 2 (A_aft − A_fore)/A_fore (holding_the_correlation_stops_at_potential_flow).

Be clear about what this does and does not do. A boattail’s read at its own length is never clipped, wherever it sits; that is the correlation as published, and a genuinely long boattail reads the same near-Mach-1 branch with no bound at all. A 4° boattail to 0.6 of the radius reads 1.29 times Munk’s line at Mach 1.5, and hpr flies it. Only the length the 16° hold invents is capped. The bound bites when the aft radius is under about 1 − √(M² − 1)/1.11 of the fore radius: two fifths at Mach 1.2, a quarter at 1.3, a twentieth at 1.45, nothing much above Mach 1.49, and only where the method’s table has started, which on such shapes it barely has.

How much that is worth is measured rather than argued, by sweeping boattails of 16° to 53.6° narrowing to between a thousandth and three tenths of the fore radius, and reading their shares back out of the table (what_the_potential_flow_bound_reaches). Two things come out.

• At the table’s rows a boattail never takes off more lift than potential flow, except in the sliver described below, where its own read already passes it and the bound never clips that.
• The bound moves a printed coefficient by about 0.060 per radian at most, at 53.5° narrowing to a thousandth of the radius. That is the printed number, after the join’s weight; the holdback on the boattail’s own cross-section is up to about six times larger near the join, where the join is barely open. 53.5° is the steepest boattail the sweep found the method willing to table at all; it refuses 53.6°, and refuses shallower angles than that where the boattail narrows less.

No committed design comes near: the steepest is Calisto’s 18.4°, whose aft radius is 0.685 of its fore radius, where the bound would need under 0.40 even at Mach 1.2.

Two caveats, both small and both real. The bound applies where the shares are computed, at the table’s rows; between rows the table interpolates, so a printed value beside a row of the next kind can sit past potential flow, by up to about 0.003 per radian on the shapes swept. And in a sliver just above the hold’s own angle (the sweep finds it at 16°, 16.5°, 17° and 17.25°), a boattail is deep enough that its own read already passes potential flow, and since the bound never clips a boattail’s own length, the hold does nothing there and the rocket flies an extrapolation nothing measured checks. That window runs a few hundredths of a Mach from where the table starts, so the join is barely open across it, and it closes as the angle or the speed rises rather than at a fixed angle.

How far to trust the 16°. It is Cubbage’s, measured at Mach 0.6 to 1.28 and on drag, and it is used here on the normal force from Mach 1.2 up. A shoulder turns the flow through a Prandtl–Meyer expansion faster than sound, where separation is less likely than transonically, so 16° is if anything early. No measurement of a steep boattail’s supersonic normal force exists to check it. Note too that the two models take opposite consequences from the same angle: separation lowers a boattail’s pressure drag toward the base value, and here it holds the lift the boattail takes off instead of letting it shrink. The argument for that is the stability one above, not a flow one. The angle is also read on each narrowing part’s own chord angle, while the drag merges adjacent narrowing parts into one cone before grading, so a boattail drawn in several parts can be graded differently by the two.

A worked example. The Arcas Robin’s boattail narrows from 2.25 in to 1.308 in over 1.757 in, so L_B/D = 0.781 and 1 − (D_B/D)² = 0.662. At Mach 2.3, √(M² − 1) = 2.071 and Fig. 5’s argument is 2.071/0.781 = 2.652, where the curve reads −0.01263 per degree, −0.7237 per radian; times 0.662 that is −0.479 per radian on the body’s cross-section. Slender-body theory gives −1.324, footnote 8 −0.103. Across the tunnel’s Mach numbers:

Their models were conical boattails of 4° to 9.5°, 0.82 to 1.18 diameters long, narrowing to 0.72 to 0.86 of the diameter; the Arcas Robin’s is steeper (15°), shorter (0.78) and narrower (0.58), so its row is an extrapolation. Their points scatter about the curve by up to about 15%. Past the curve’s end (an argument of 6.1: a short boattail at a high Mach number) hpr holds its last value. BodyModel keeps footnote 8 for comparison.

A table. One run of the method takes a few milliseconds, too slow for every step of a flight. So the first time a flow faster than Mach 1.2 needs it, hpr runs the method every 0.05 in Mach from Mach 5 down, to the lowest Mach at which it holds, and keeps the results. Between those Mach numbers it interpolates in a straight line. That took about 0.3 s once per rocket in a debug build on the development Mac (measured by hand, for a body the method takes from Mach 1.2); a body whose join starts higher adds about 48 runs for the bisection below. A rocket that never passes Mach 1.2 never pays it.

The join. Write SB for slender-body theory, SE for the shock-expansion method, and M_j for where the join starts: Mach 1.2, or the lowest Mach at which the method holds if that is higher. hpr narrows that Mach down between two rows of the table by halving the gap (bisection) until no smaller step exists in the computer’s numbers. It then runs the method at that Mach and adds the result as an extra row. The start must be that exact: there the method’s shares climb from zero like the square root of the distance in Mach, so a start off by δ puts √δ-sized shares in that row. So the start moves smoothly with the nose’s shape instead of in 0.05 steps. A cone with a 20° half-angle (the angle between its side and its axis) joins from Mach 1.341910; each 0.1° steeper, up to 20.5°, moves the start about 0.0027 later, to 1.355500. From M_j to M_j + 0.3, each covered part’s slope, moment and station move in a straight line from SB’s to SE’s:

C_Nα = C_Nα,SB + w (C_Nα,SE − C_Nα,SB), w = (M − M_j)/0.3, clamped to 0 to 1.

Every piece is a straight line in Mach, so nothing jumps. The test the_supersonic_join_has_no_jump looks at ±1e-9 in Mach on each side of the join’s ends, of table rows, between rows and at Mach 4.999, and a_blunter_cone_joins_where_the_method_starts_to_hold does the same for the 20° cone, whose join starts higher. the_joins_start_moves_with_the_nose_not_in_steps pins that cone’s start and the 20.5° cone’s to 1e-6, checks the start is off the grid, that the cylinder’s share at the start is under 1e-5 per radian (about 1e-7), and that the start moves by less than 1e-7 when the cone steepens by a millionth of a degree (2.7e-8). a_boattailed_body_flies_the_method_without_a_jump does the same ±1e-9 probe for the finned rocket with its boattail and tail tube. Mach 1.2 to 1.5 is a judgement: below Mach 1.2 the flow over the nose is transonic, which the method doesn’t cover, and Mach 1.5 is the lowest Mach at which NASA measured the Arcas Robin (ADR-034, the decision behind it). Small changes in shape can still switch a body between the two models, for example a nose so steep that the method never holds at any Mach up to 5, so the rocket keeps slender-body theory throughout (issue #87).

A worked example: the Arcas Robin’s body. NASA measured the Arcas Robin’s body without fins in a wind tunnel (TN D-4014). Flown through a flight’s own code, with the secant-ogive nose (a tangent ogive’s cousin, its arc larger) fitted to the report’s coordinates and the boattail left off, the nose and cylinder give the method’s own values: equal on the table’s rows (Mach 1.5, 1.8, 2.3), within 1e-4 between them.

With the boattail on (the lip behind it off, a flare the method doesn’t take), the boattail takes its measured share, 0.31 to 0.62 per radian off, most at Mach 1.5 (footnote 8 took 0.03 to 0.18). At α → 0 the short model then reads 12.0% to 26.2% below the measured line and the long model 29.7% to 33.3% below it. That line is fitted over the plotted angles, so it carries crossflow lift and these columns leave body lift out: fitted the same way, with the body lift a flight adds, the same body reads 3.4% to 41.0% high (Checking the shock-expansion method).

hpr’s committed Arcas Robin design flies the method to its base since Blunt tips and A lip in a boattail’s wake: the last two columns are the method’s own values for its power-series nose, cylinder and boattail, with the lip carrying nothing. Slopes are per radian on the body’s cross-section, at α → 0; the measured slope is fitted over the plotted angles with the boattail and lip on, so it also carries some crossflow lift and their share, which is why every column reads below it here.

The rows are in validation/fixtures/aero/shock-expansion.json (arcas_robin: nose_and_cylinder, in_flight, in_flight_with_boattail and as_designed), written by cargo xtask aero; no test re-reads the flight columns, so they are regenerated by hand.

What it leaves out. Steps, and widening shapes that are not cones, as above; a conical flare has its own section (A flare through the method); a blunt or vertical tip takes the cap of Blunt tips, below. The method itself has no crossflow lift; a flight adds body lift on top.

A lip in a boattail’s wake

What this covers: a short flare at the very base, behind a boattail, like the reflex lip of NASA’s Arcas Robin models. How far to trust it: hpr gives such a lip no normal force faster than sound, which is what the measured pitching moment supports, but the moment bounds the lip rather than measuring it.

The rule. A lip that sits wholly in a boattail’s wake carries no potential-flow slope from the Mach number where the method takes over; below the join it keeps slender-body theory’s 2 ΔA/A_ref, and the join blends the two, so nothing jumps. hpr takes the shelter’s share from its drag model (ADR-030, which takes the same lip’s drag away): wholly in the wake up to a rise of a quarter of the boattail’s drop in diameter, not at all from half of it, and the wake fades over any tube between them. On top of that fraction the normal force asks one thing the drag model doesn’t: the lip must be no longer than the boattail’s drop in diameter, the wake’s own scale, or it grows out of the wake however little it rises. The decision record on the lip, ADR-039, sets out the readings behind the share itself.

A lip part way out of the wake. Where the wake covers the lip only partly, the drag model has always graded it so. Since M1.8e10 the normal force reads the same number as a weight: the method’s share counts for the wake’s share of the lip, slender-body theory for the rest, exactly as they blend across the Mach join (ADR-041). It is the whole wake fraction, not the rise alone: the shelter fades with the lip’s rise, with any tube between it and the boattail, and with anything else in the way.

Be clear about what that buys. The jump is gone: a lip drawn a hair taller no longer switches the whole body between the two models, which it used to do by a third of the normal force and 1.77 calibres of center of pressure on the tests’ rocket at Mach 3. The sensitivity is not gone; it is spread over the band. On that rocket, at Mach 3 and 4°:

That table is a design sensitivity: how the printed answer moves as you draw the lip differently. It is not the same as how far apart the two models are, which is what a lip in the band is actually uncertain by. At one fixed shape (a lip rising a quarter of the drop) the method and slender-body theory differ by 33.0% of the normal force and 1.77 calibres, and nothing measured says which is right for a part-sheltered lip. That number has not changed; what changed is that a rocket can no longer cross it in a ten-thousandth of its geometry.

Why nothing. Three readings point the same way.

• The tunnel itself. TN D-4014 ([D4014] p. 6) traces an odd chamber axial force at Mach 1.50 and 1.80, fins off, to the reflex lip, and says the effect is “masked” once separation runs over the boattail at higher Mach numbers or the fins thicken the boundary layer, and that the longer model shows none of it, “probably because of the thicker boundary layer at the model base”. So the lip does something at those two speeds, and those are the very rows whose moment implies a negative share below; what it does there isn’t a normal force this model can carry.
• Seiff’s own limits. His embedded Newtonian flare method holds for “thin shock layers when the flow is not extensively separated” ([S62] p. 13), and he notes that “a 90° ramp will invariably separate the flow” (p. 4). The Arcas lip’s face stands about 57° to the axis, behind a 15° expansion.
• The size, at most. Taking Seiff’s method anyway as an upper bound (eq. 9, p. 12, which for a conical flare at one dynamic pressure is 2 (q₁/q∞) cos²θ ΔA/A_ref, with q₁ the flow that has expanded through the boattail’s turn, and θ taken to the axis, 56.8°, where Seiff measures it from the local stream, the looser of the two) gives 0.044 per radian at Mach 1.5 falling to 0.014 at 4.63, against slender-body theory’s 0.178 at every speed.

What the moment says, and what it can’t. For each fins-off row, the share at the lip’s station that would put hpr’s center of pressure on the measured one runs from −0.256 ± 0.068 per radian (short model, Mach 1.5) to +0.229 ± 0.084 (long, Mach 3.96), changing sign with Mach number and with the model’s length. Fitting one share:

A χ² per degree of freedom of 1 means rows agreeing within their own error bars. The short model’s 6.4 means its rows disagree among themselves; the long model’s 0.9 means its five agree, on a share of +0.108, five times Seiff’s bound at that speed and three standard errors above zero, yet still two below slender-body theory’s.

So the moment does not settle the lip, and the model doesn’t rest on it. The reason is in how the number is made: it blames the lip for every miss in the center of pressure, and hpr’s body alone reads 15% to 19% high on the long model at Mach 1.8 and 2.3, which shifts its center of pressure by far more than any lip. The short model’s own fit comes out negative, which no flare can produce. What the moment does say is that slender-body theory’s 0.178 at the base is too much: eight standard errors out on the short model, two on the long.

How it was checked. The committed designs, fins off, through a flight’s path, fitted at the tunnel’s plotted angles as Checking the shock-expansion method fits them. Slopes are per radian on the body’s cross-section; centers of pressure are calibres aft of the nose tip. The last column is where hpr’s whole-body center of pressure would sit if the lip carried slender-body theory’s share instead of nothing. hpr’s slopes here are fitted over the tunnel’s angles, so they carry body lift; the same bodies’ slopes at α → 0, in Checking the shock-expansion method’s table, are lower.

The rows are in validation/fixtures/aero/arcas-robin-lip.json, written by cargo xtask aero, and a test holds this table to it cell by cell. With the lip left off entirely the same bodies read within 0.05 percentage points of these rows at ten of the eleven, and 0.5 at Mach 1.5 on the short model, so the lip changes little but which parts of the body the method may cover.

What it means for a rocket. Most rockets have no lip, and nothing changes for them. For one that does, the rule decides whether the whole body flies the method or slender-body theory, so it is worth more than the lip itself: on the Arcas Robin’s short model at Mach 2.96 the body’s slope goes from 2.08 per radian to 3.84, and the whole rocket’s from −16.3% against the tunnel to +3.7% (Normal force through Mach 1). The extra lift sits on the body, ahead of the fins, so the center of pressure moves forward by 0.22 calibres there: a little less stability margin, and a good deal more restoring force.

What it leaves out. The lip still has drag, and its own wake rule there (ADR-030). Nothing here measures a lip’s lift directly: the tunnel gives forces for the whole body, and the moment bounds the share rather than measuring it. The shelter used to be a switch in shape, of the family issue #87 tracks, and a large one, because it decides whether the whole body flies the method. Since M1.8e10 the rise no longer switches it: on the test rocket at Mach 3 and 4°, lips rising 0.2499 and 0.2501 of the boattail’s drop now agree to a ten-thousandth. What is left is how far apart the models are at one shape: 33.0% of the normal force, and the center of pressure 1.77 calibres forward on slender-body theory, spread over the band the wake grades (above). A narrowing part behind the run is a boattail the method hasn’t covered, not a lip, and keeps slender-body theory’s share. A widening part behind a boattail that is too long for the wake stays a lip as well, and still takes the whole body off the method: the flow reaching its corner is the wake’s, which the march does not compute, so A flare through the method does not apply to it.

Where a flare’s march stops

In short: a flare is a transition that widens toward the tail, and the method will march one, but only up to a limit, and that limit is not where the flare’s shock detaches. It is where the corner’s turn would take the flow to Mach 1, which is a property of hpr’s method rather than of the air, and it depends on the whole body ahead of the flare: on the body measured below it falls short of a wedge’s detachment angle at Mach 1.5 and runs past it at Mach 2, and taking the tube away moves it past the wedge at both. From Mach 2.13 to Mach 5, the highest checked, the limit is neither: it is the 30° where the cone tables end. The flare’s own detachment angle is not known here: the wedge’s is a conservative stand-in for it. This section is the measurement, and it is why the attachment test a flight uses had to be chosen rather than read off the march’s refusal (ADR-045). What a flight does with a flare is the next section, A flare through the method, and how close that comes to a measured flare is the one after it, What a marched flare is worth. Neither this section’s edge nor the drawn-out reading past it is itself compared with a measurement.

Second-order shock-expansion turns every corner isentropically (no entropy rise, so no shock) with the Prandtl–Meyer angle ν ([SD56]) eq. 3. A flare is a compression corner, so the turn spends ν: the march goes on only while the flow reaching the flare has enough of it left to turn through the flare’s angle without dropping to Mach 1. Whether a shock instead stands attached to that corner is a separate question, and it is the one that decides whether the march is modeling the real flow at all.

The table below sweeps the flare’s angle on a fixed body (a pointed 2.75° cone, five calibres of tube, and a conical flare) and bisects, until the two angles are adjacent double-precision numbers, the steepest flare the march accepts. The two angles are NASA TN D-4865 model 2’s; the layout is not. That model is blunt-nosed and has no tube at all, and the edge depends on what is ahead of the flare, because that is what sets the flow reaching it. Angles are quoted to seven decimals so the differences add up, and the detachment column is taken at the free-stream Mach number (the flow reaching the flare is a little faster, which would move the wedge’s angle by about 0.02°). Both tests are in shock_expansion.rs: a_flare_marches_to_the_isentropic_turn_not_to_detachment and past_mach_2_13_the_flare_stops_where_the_cone_tables_do.

How much of that is the tube. A great deal, and it is the point rather than a caveat: what the march has left to spend is ν of the flow arriving at the corner, and the body ahead sets that flow. Keeping the same cone and flare and changing only the tube’s length:

So the first row of the first table (the march stopping short of a wedge’s detachment) is a property of that five-calibre body, not of the method: on the report’s own tube-less layout the Mach 1.5 edge is 14.19°, two degrees past the wedge’s limit. What does not depend on the body is the conclusion: the march’s edge is set by the corner’s isentropic turn, and it lands on both sides of a wedge’s detachment angle depending on nothing more than how long the tube is.

The detachment angles are a wedge’s largest deflection ([R1135], through wedge_detachment_angle_rad, which lives with the blunt-tip cap because that cap uses the same relation). A cone’s shock holds to steeper angles than a wedge’s, and a conical flare on a cylinder sits between the two, so the wedge’s column is a conservative stand-in, not the flare’s own boundary. Only the rows where the march stops below the wedge’s angle prove anything about attachment; where the march runs past it, the flare’s own limit may still be higher. Which boundary a flight uses is the next section’s subject. What the table shows is that the march’s edge lands on both sides of any such boundary: you cannot tell, from hpr returning an answer, that the flow it modeled is the flow that would be there.

The last two rows of the first table are a different limit altogether. Each element’s tangent cone is looked up in NASA SP-3007 Table 2, whose slopes stop at 30° (the milestone that took them there from 24° is M1.8e11 (ADR-042)), so from Mach 2.129702032593 to Mach 5, the highest checked, every Mach number gives the same edge. The reference data runs out before the flow does, and the last column then says nothing about attachment.

An 18.5° flare, the report’s angle, on the 2.75°-cone body above (not the flared rocket of the next section, whose numbers are close but not these). The march accepts it from Mach 1.721760; a wedge’s shock reaches 18.5° only at Mach 1.767575. Between the two, hpr returns a number for a flare whose shock is, on that reckoning, detached: a bow shock standing ahead of the juncture with a pocket of subsonic flow behind it, which an isentropic corner turn does not describe. TN D-4865’s lowest run, Mach 1.50, is below both, and there the march refuses outright, as the report itself says it should. On a shorter body those two Mach numbers move, as the table above shows.

What it leaves out. These digits pin what this program does, not what air does: every one of them comes from bisecting hpr’s own refusal, and the 30° rows come from where a lookup table ends. A band of very shallow flares is not marched by the second-order law at all: the pressure behind such a corner moves away from its tangent cone’s rather than toward it, so its one element is reduced to the older generalized method, which since M1.8e19 is read rather than refused (A near-flat flare). Where that band sits is a property of the body ahead of the corner, and it moves by orders of magnitude: 0.773° to 0.823° at Mach 3 on this 2.5-calibre body, against 0.0066° to 0.0081° at the same Mach number on the flared rocket of A near-flat flare. The whole edge is inviscid, too. From Mach 2.96 up TN D-4865 records the boundary layer separating ahead of the flare and reattaching on it, which moves the pressure rise downstream of where a tangent body puts it; nothing here models that. And below Mach 1.5 nothing here was measured, although a flight uses the method from Mach 1.2.

A flare through the method

In short: since M1.8e17 a rocket with a flare flies the shock-expansion method rather than dropping to slender-body theory the moment one is drawn. The method marches through the flare’s corner while the shock there stays attached; a steeper flare is read as one of the same radii drawn out to the steepest attached angle, which is what keeps the answer from jumping as a shape or a speed crosses that boundary. How far to trust it: the attachment test is a standard relation, applied at a corner it was not derived for; the reading either side of the boundary it draws agrees to four parts in 1e11, though its slope kinks there; and what it is worth against the one measured flare in the sources is the section after this one: −1.9% and +7.0% at Mach 1.9 and 2.3, +13.4% at 2.96, and +51.5% and +50.4% at 3.95 and 4.63. Read the numbers below as what this program does, and that section as how close it lands. What qualifies is narrow: a conical flare, flush with the part ahead of it, not in a boattail’s wake, with nothing behind it that carries lift of its own. A boattail then a flare is a lip in a wake and keeps that rule; any other widening shape still ends the run. The decision record is ADR-047.

Why a test had to be picked. The march will return a number for a flare whose shock has long since detached (the section above measures exactly that), so “the method answered” is not evidence the flow it modeled is the flow that would be there. Something independent has to say where the corner’s shock detaches.

The test. A flare’s shock springs from a circular corner, not from a point apex. Where the shock forms, the flow is two-dimensional: the body’s radius is the scale over which the axisymmetric relief acts, and at the corner itself none of it has happened yet. So hpr uses NACA Report 1135’s ([R1135]) largest deflection behind an attached plane oblique shock (eq. 168 into eq. 138), the same relation, in the same function (wedge_detachment_angle_rad), that TN D-4865 ([J68]) p. 5 uses to hand a blunt tip’s cap over to this method. One attachment rule, at both corners the program has.

Two details matter.

• It is read at the flow reaching the corner, not at the free stream. The body ahead has already changed the air, and which way depends on what that body is: the ogive nose and tube of the tests’ flared rocket leave it a little slower than the free stream, while the 2.75° cone and tube of the section above leave it a little faster. The march knows by how much (aft_flow). On the tests’ flared rocket a Mach 2.0 free stream reaches the corner at Mach 1.9998, so the limit is 22.9698° and not the free stream’s 22.9735°. The march is downstream-only, so the flare cannot change the flow arriving at its own corner, which is what lets the limit be worked out before the flare is drawn.
• The cone tables cap it at 30°, because past that the march has no tangent cone to relax toward (tangent cone, ADR-042), but that bounds the flare’s surface angle, while the shock bounds the turn at its corner, so each caps its own quantity. On a flare behind a cylinder, where the surface ahead is at 0°, the two are the same number and the cap binds from Mach 2.5192034260 of the flow reaching the corner up; a faster flow buys nothing above that.

Because the shock’s bound is on the turn and not on the flare’s angle, a flare put straight onto an ogive nose (no tube between them) is read differently from the same flare behind a tube: the corner there turns the flow by the flare’s angle less the nose’s base slope, so the limit bites at a steeper flare. On the tests’ flared rocket (an ogive nose 0.25 m long on a 27 mm radius, then a 0.7 m tube of that radius) the surface ahead is at 0° and the turn and the angle are one number:

A cone’s shock holds to steeper angles than a wedge’s, so if a conical flare on a cylinder does sit between the two, this errs one way only: it stops reading some flares whose shock is in fact still attached, and reads none whose shock is not. Nothing here measures where a flare’s own boundary actually is, so that “if” is an argument, not a result, and erring low is not the same as erring safely. A flare just past the limit is not left unread; it is read as a different flare, and how different nothing here measures either.

Past the limit, the flare is drawn out. A flare steeper than the limit is read as a flare of the same radii stretched to the limiting angle, longer and shallower, turning the same air through a corner the shock can hold, with its center of pressure put back on the real flare, at the same fraction along it. Because the radii are kept and the angle is not, every flare steeper than the limit reads the same force: held at the limit, in other words. There is no bound on how far that goes, and it under-reads badly at the extreme: at Mach 2 a 75° flare, an annular face a detached bow shock would stand in front of, reads 1.638 per radian against slender-body theory’s 2.010, where the truth is above both. It reads a steep flare as less stabilizing than it is, which is the safe direction for a stability margin and the wrong one for a load. This is the boattail rule turned around: a boattail past 16° reads the correlation of one of the same radii drawn out to 16° (ADR-040, A boattail faster than sound). The radii are what set how much air the flare turns, and they are never changed.

That is also what makes the answer continuous, by construction rather than by tuning: at the limit the drawn-out flare is the real flare, so the two readings are the same body. The test nothing_jumps_where_the_flares_shock_detaches probes either side of the boundary. At Mach 2.0 (a row of the table, so the reading is that row’s and not an interpolation) the boundary is a flare of 22.969761173077°:

A hundredfold smaller probe moves the answer a hundredfold less, so what the probe finds is a slope and not a step: the reading is continuous across the boundary. Its slope is not, and nothing here claims otherwise: a cap makes a kink, because below the boundary the flare’s angle moves the body the march sees and above it only the radii do. Measured at the same place, dC_Nα/dδ changes by −31.4% across the boundary on the whole rocket and by −141.6% on the flare’s own share, where it changes sign. The marched branch is not smooth in the angle either: the same probe at 20°, away from any boundary, finds +3.5% and +41.1% (the_cap_makes_a_kink_in_the_slope_even_though_the_reading_holds). Probing the Mach number instead, at the 18.5° of TN D-4865’s model 2, whose shock holds on the tests’ flared rocket from Mach 1.767666917849, gives 3.622e-10, 3.622e-8 and 3.622e-6 for the same three probes. That half is taken on the table’s own rows rather than on a reading between them: the crossing falls inside the Mach 1.75 to 1.80 interval, where the reading is a straight line between rows and would look continuous whatever the two branches did.

What it is worth. The tests’ flared rocket is an ogive nose 0.25 m long on a 27 mm radius, a 0.7 m tube of that radius, a 10° conical flare 0.3 m long opening to a 79.9 mm radius, a 0.2 m tail tube and four fins. Its reference is the largest diameter, 0.1598 m, and the numbers are at a small angle of attack. The method reads less normal force than slender-body theory and puts the center of pressure forward of it, so this rocket now reads less stable rather than more:

So for a flare of this size the stability margin drops by a tenth to a quarter of a calibre, growing with Mach number. a_flared_body_flies_the_method pins every figure in the table.

The old behavior is still selectable, for comparing: BodyModel::with_supersonic_flare(SupersonicFlare::SlenderBody) in Rust, or {"supersonic_flare": "slender_body"} in the body model’s JSON, reproduces every number from before this milestone. Which to fly is not settled here: neither column has been compared with a measured flare, and the method is the default because it is the model the rest of the body already uses faster than sound, not because it is known to be closer.

What it leaves out.

• The one measured flare is not this flare. What a marched flare is worth compares an 18.5° flare on a 2.75° cone, at six speeds, three of them with its boundary layer separated. The 10° flare in the table above is not that body, so the size of the difference it shows is still a change of model rather than a measured correction.
• The flare’s own detachment angle is still the wedge’s. A conical flare on a cylinder sits between a wedge and a cone, and nothing here measures where it actually is.
• Below about Mach 1.5552 the method has no reading for an 18.5° flare on the tests’ rocket at all. The corner’s isentropic turn runs out before its shock detaches (the section above), so the body’s table of the method’s shares starts there and the join carries the reading up from slender-body theory’s over 0.3 Mach. That is continuous (a ±1e-9 probe at the join’s start moves the slope 8.5e-10 of itself), but it means a flare’s march is not used at all at the low end of what a flight uses.
• A near-flat flare is read by the older, rougher method. Between about 0.0004° and 0.059° on the tests’ rocket, depending on the Mach number, a flare’s single element is reduced: the pressure behind its corner sits just past its tangent cone’s while the gradient the tube delivers still pushes it away, so hpr reads that element by the generalized shock-expansion method instead (issue #81: the method’s limit near a sharp tip is the same reading on a nose). Until M1.8e19: the near-flat flare the march refused such an element behind the nose and the whole body fell back to slender-body theory at every Mach number, which was a switch worth −8.3% and 1.16 calibres. A near-flat flare below solves for where the region is at each Mach number, says what the change was worth, and gives the one step that is left: +0.129% and 0.0051 calibres on this rocket, and up to +4.3% and 0.19 calibres on a body with a short shoulder.
• The march ends at the flare, and nothing behind it may carry lift. Anything behind the flare takes slender-body theory’s share, which for a tube is nothing, but a part that carries a share of its own, such as a small tail cone behind the flare, takes the whole rocket off the method again, at every Mach number, exactly as a second flare would. A flare followed by a plain tube and fins is the layout that flies. Where it does fly, the relaxation the method would give the tube behind the flare is left out, which reads a little less stable, not more.
• Everything here is inviscid. From Mach 2.96 up TN D-4865 records the boundary layer separating ahead of the juncture and reattaching on the flare, which moves the pressure rise downstream of where a tangent body puts it. Nothing here models that.

What a marched flare is worth

In short: A flare through the method, above, says what hpr does with a flare: it marches the shock-expansion method through the flare’s corner while the shock there stays attached. This section says how close that comes to a measured flare, which before M1.8e18, the milestone that did this work, nothing had.

There is one flared body in the sources whose normal force and pitching moment are printed: NASA TN D-4865’s model 2 ([J68]), a blunt 2.75° cone with an 18.5° flare, in the Langley Unitary Plan tunnel from Mach 1.50 to 4.63. Against it hpr reads the normal-force slope −1.9% at Mach 1.90, +7.0% at 2.30, +13.4% at 2.96, then +51.5% and +50.4% at 3.95 and 4.63, with the center of pressure within 0.05 calibres through Mach 2.96 and 0.088 at 3.95. The report’s own shadowgraphs show that flare’s boundary layer separated ahead of the juncture from Mach 2.96 up, but the cost only shows in the two fastest rows; why is not settled here. Below Mach 1.5289 there is no reading at all, and a flared body falls back to slender-body theory instead.

One body, one flare angle, six speeds, three of them separated: that is the whole of the evidence, and it is not enough to call the model right, only enough to say where it is not obviously wrong. The decision record is ADR-048.

The body. Two things about it bear on the comparison: its nose is blunter and more compound than model 1’s, so hpr had to learn to hand a blunt tip’s cap over on a later piece of a nose; and the report’s own drawing does not quite close, so which of its printed numbers to keep had to be chosen. Neither is worth much (the closure is worth 0.64 points at most, measured three ways), but both are choices, so here they are.

Fig. 3(b) (printed p. 91) draws model 2 in base diameters, d = 0.583 ft (0.178 m). Its nose is not a sphere-cone: a 0.257 sphere from the tip, then a second arc of 0.429 whose center sits 0.135 below the axis, then the 2.75° cone, then the 18.5° flare. Those three printed radii fix everything else. The two arcs are internally tangent, so their centers are 0.429 − 0.257 = 0.172 apart, which with the 0.135 offset puts the second center 0.3635786 aft of the tip; its tangent to the 2.75° cone then falls at 0.3429960 aft of the tip, matching the drawing’s printed 0.343 to four figures, at diameter 0.5870, which misses its printed 0.586 by 0.001 of a diameter. A fourth printed dimension checks the same derivation from the other end: 0.722 runs from the arc’s center to the flare juncture, and 0.3635786 + 0.722 = 1.08558 against 0.343 + 0.743 = 1.08600, a residual of 0.0004. hpr draws that blend arc as a circular arc (the tangent ogive’s shape with its radius ratio solved for a 0.429 arc), and the profile it builds misses the drawn circle by 8.3e−17 of a diameter.

The printed dimensions do not quite close, and the gap is the flare’s. Nose, cone and flare come to 0.3429960 + 0.743 + 0.523 = 1.6090, the printed length exactly. The cone closes on its own numbers too: 0.586 + 2 × 0.743 tan 2.75° = 0.65737, against the printed 0.657. The flare does not: 0.657 + 2 × 0.523 tan 18.5° = 1.0074, against a base that is 1.000 by definition, and 1.0084 with the nose taken from its three radii rather than from its printed 0.586. So either the flare is shorter than 0.523, or it is shallower than 18.5°.

hpr keeps both half-angles, because they are the report’s text and not only its drawing: “a blunted cone with a 2.75° half-angle and a flare afterbody having an 18.500° half-angle” (printed p. 8), stated to three decimals, along with the nose, the length 1.609 and the base 1.000, which are the numbers the measured coefficients are divided by. What gives is the split of the length between the cone and the flare: a 0.7576199 cone and a 0.5083841 flare, which puts the juncture 0.0146 diameters aft of the printed one, at diameter 0.6598 against 0.657.

Two other closures are computed and published beside it, so the choice can be checked rather than trusted. Keeping the printed lengths and scaling the whole body to a 1.000 base moves the error by −0.12 to −0.17 points (points here and below are percentage points of error). Keeping the printed lengths and the base and giving up the flare’s stated angle instead (18.5° becomes 18.0864°), moves it by +0.06 to +0.64 points. Over all three the spread is 0.64 points or less in the slope and 0.0043 calibres or less in the center of pressure, and Mach 1.50 is refused in every one of them, so nothing below turns on which closure is flown.

What it is compared with. Fig. 8(b) (printed p. 102) plots normal force C_N, pitching moment C_m and axial force C_A (all three as coefficients, C_m about the nose tip on the body’s own length) against α at 0°, 4°, 8° and 12°, for each of six Mach numbers. The circles are the report’s experiment: the surface pressures of its tables VII to XII integrated over the forebody (printed p. 12), so no base pressure is in them. Every circle was read off the page’s 300-ppi scan by pixel analysis, the method written for model 1 in M1.8e7; the α = 0 circles come out at −0.0039 to +0.0043 where they should read 0, which is what the plotting itself is worth. C_A is not read, because hpr’s supersonic drag is a separate model this comparison does not touch. hpr’s slope and center of pressure are fitted the way that section fits model 1: a straight line through hpr’s own C_N at those same four angles, body lift included, so the two sides are the same quantity.

A positive number in the last column means hpr puts the center of pressure further aft than the tunnel did, which reads as more stable than the rocket is; a negative one reads as less. The half-calibre the rest of this page uses as a target is the scale to hold them against.

The flare is most of what is being compared. Its own share of the body’s slope is 52.1% at Mach 1.90 and 60.8% at 4.63, and never below 51.5% (at Mach 2.30) in between. That share acts 1.382 to 1.390 calibres aft of the tip, on the flare itself, which runs from 1.101 to 1.609. So this is a test of the flare and not of a body that happens to have one.

How much of the miss is the flare’s. The same report, the same tunnel, the same figure and the same reading and fit also give model 1, a sphere-cone with no flare, the body Blunt tips already checks. Putting the two side by side separates what the flare costs from what the rest of the body costs:

They are not one body with and without a flare: model 1 is an 11.5° cone on a 0.175-diameter nose radius, 1.755 diameters long, and model 2 a 2.75° cone on a 0.257-diameter one, 1.609 long, so the flare adds column bounds what the flare costs rather than measuring it. Read it as a signed difference and nothing more: it is not a verdict on which body hpr reads better. At Mach 1.90 the unflared model 1 is almost exact (+0.011%) and the flared one is 1.9% low, so in size of error the flare is the worse row, not the better one.

Taken that way the column still says something clear. Through Mach 2.96 the flare moves the error by −1.9 to +0.9 points; that is, by less than the rest of the body already misses by. At Mach 3.95 and 4.63 it moves it by 21.7 and 18.3 points, an order of magnitude more, and that is where the measured C_Nα itself falls away (from 1.594 at Mach 2.96 to 1.270 at 3.95) while both attached-flow methods on the figure, the report’s own and hpr’s, stay between 1.57 and 1.96.

The obvious explanation is the flow: from Mach 2.96 the report’s shadowgraphs show the laminar boundary layer separating ahead of the juncture and reattaching behind it (printed p. 10), and nothing in an attached-flow method describes that. How far the report backs it up is worth being exact about. It blames the separated flow for its own method’s disagreement with the measured pressures at the high Mach numbers (printed p. 10) and for its over-prediction of axial force (printed p. 12). It says nothing at all about what separation does to the normal force or the pitching moment. Nor does its method carry a separation signature on this body: it reads +24.8% at Mach 1.90 and +31.6% at 2.30, both attached rows, against +28.8% and +20.3% at the separated ones, uniformly high on model 2 and near the tunnel on model 1, which is a large offset this milestone has not explained. So read the separation as consistent with the two fast rows rather than measured by them. What nothing here settles is why it costs 0.9 points at Mach 2.96, where the report says it has already begun, and twenty times that at 3.95.

Where the method has no reading. At Mach 1.50 hpr refuses model 2 outright, and the refusal is worth following, because it is not the rule the section above describes. Two limits decide what the march does with a flare’s corner: the steepest surface angle whose shock stays attached there, which is the rule that section sets, and the steepest the march itself can turn the flow through. Both are in the table below as surface angles of the flare, so they can be read against its 18.5° and against each other:

The turn the corner actually makes is each of those angles less the 2.75° of the cone ahead. From Mach 2.3 up the third column is clipped at the cone tables’ 30° (past that an element has no tangent cone to relax toward), and the search behind the fourth stops at the same place, so neither is a measured edge there and the fourth says so rather than printing the cap as though it were one. At Mach 1.50 the flare is steeper than the 15.4885° its corner’s shock holds, so ADR-047’s rule draws it out to 15.4885°, and the march then refuses that body too, because the steepest flare it can march there is 15.3647°, a tenth of a degree shallower. The two limits are different things: one is where the corner’s shock detaches, the other where the corner’s isentropic turn runs out (Where a flare’s march stops, ADR-045), and which is the tighter one changes with speed. Below where they cross, drawing a flare out to the shock’s limit lands past what the march can do. Swept every 0.005 Mach from 1.05 to 4.63 the reading turns on exactly once, and bisecting that one crossing to f64 resolution puts hpr’s first reading of model 2 at Mach 1.5288696; just below it the two limits agree to five parts in 1e14, both 16.2844275°, so the reading begins exactly where they cross. (The fixture keeps the whole f64; seven figures is what the three operating systems CI runs agree on, since each regenerates the bisection to within a part in 1e12 of the others.) That is a different number from the Mach 1.5552 the section above quotes for the same 18.5° flare, and it should differ: the body ahead of the corner is different, so the flow it delivers there is different, and the crossing moves with that flow. Below that a flared body takes slender-body theory instead, carried up over 0.3 Mach by the join, which is continuous, but means the method is not used at the low end of what a flight flies. The report says the same thing about its own method at that speed: “at M∞ = 1.50, the shock wave produced by the flare is not theoretically attached” (printed p. 10), and its method reads +28.6% there against +2.0% on model 1.

The rows above are in validation/fixtures/aero/marched-flare.json, written by cargo xtask aero; the readings in tn-d-4865-flared-cone.json, with how each circle was read. A test holds every table here to the fixture, cell by cell. The reading is hpr’s own rule and not a second copy of it: hpr-design has no spherical-cap nose, so model 2 cannot be flown through a Rocket, and the_flare_is_read_as_the_model_reads_it pins the fixture’s reading against the model’s own shares: share by share, to a part in 1e12, on a flared body the design route can express, at angles the corner’s shock holds and at angles it does not. That pins the arithmetic, not the physics: an error in the rule itself would be in both and pass.

What it leaves out.

• Three of the six rows are a separated flare. From Mach 2.96 the report’s shadowgraphs show the boundary layer separating ahead of the juncture; nothing in hpr models that, and the +51.5% and +50.4% at Mach 3.95 and 4.63 are what those rows cost on this body, which the report’s own words make consistent with separation rather than caused by it, since it never says what separation does to the normal force. No source here says how a separated flare scales with the flare’s angle, its length or the boundary layer’s thickness, so those two numbers do not transfer to another flare.
• One body and one flare angle. 18.5°, on a 2.75° cone, at six speeds. Nothing here measures a shallow flare, a steep one, a flare behind a cylinder rather than a cone, or a flare on a pointed nose. A reader with a different flare has no measured error to apply: what this section supports is that hpr’s flare is not obviously wrong where the flow stays attached, not that it is good to 7% on some other body.
• The drawn-out reading past the limit is measured against nothing at all. The one row where it would have applied is the row the march then refused, so the rule above the corner’s limit is still a construction chosen for continuity.
• The measurement is the forebody only. Its C_N and C_m are integrated surface pressures with no base term, which is what hpr’s body model computes too, but it also means the tunnel’s own balance never weighed this body, and a reading error of about 0.004 in C_N sits under every circle.
• C_A is not compared. The same figure plots axial force, and the report notes its own method reads it high where the flare separates. hpr’s supersonic drag is a different model with its own checks; this milestone did not touch it.
• The drawing is 1% inconsistent and one closure had to be chosen. The spread over the three closures is small (0.64 points, 0.0043 calibres) and the flare’s own angle is inside it, but it is the drawing’s disagreement with itself, not an error bar on the measurement.
• No target was set, and none is met or missed here. M1.8e18 asked what a marched flare is worth, not that it reach a number. The tables above are the answer.

A near-flat flare

In short: a flare that opens by very little turns the flow so little that the second-order shock-expansion method’s own pressure curve has nothing left to describe, and the march falls back on the older generalized method for that one element. Until M1.8e19: the near-flat flare hpr refused to read such an element at all behind the nose, which took a rocket with a flare of about a third of a millimeter’s rise off the method entirely. It now reads it. This section says which flares those are (the two angles that bound them are solved from two equations about the corner’s own flow, rather than found by bisecting the model’s refusal), what the change was worth, and the one small step that is left.

How far to trust it. No wind tunnel has measured a flare this shallow, and the report does not say what it would have done here, so what follows is hpr’s own reading of the report’s own limit, chosen because it is continuous in the flare’s angle and smooth through the region, not because it is known to be nearer the air. What changed is which model runs, not how well either matches a measurement. And where the region sits depends entirely on the body ahead of the corner: a few thousandths of a degree on the rocket measured here, nearly a degree on the body of Where a flare’s march stops (a pointed 2.75° cone and five calibres of tube, whose radius is nearly four times as large).

What a flare that small does to the method. The march is the walk along the tangent body’s straight elements, from the nose tip aft, that the method makes. It fixes the pressure just behind each corner from the Prandtl–Meyer turn there, and then lets it relax along the element toward the pressure on that element’s tangent cone, as

p = p_c − (p_c − p₂) e^(−η), η = (∂p/∂s)₂ (x − x₂) ⁄ ((p_c − p₂) cos δ₂)

(NACA TN 3527 eqs. 8 and 9), where x − x₂ is the distance back from the corner. The same exponent written as a rate per meter is the k of What a crossing is, and what it costs. The symbols are the method’s own, listed under Bodies faster than sound: δ an element’s angle to the axis, p the pressure over the free stream’s, s distance along the surface, r the radius at the corner, Ω a stream tube’s widening and B = γpM²/(2(M² − 1)); θ below is the flare’s own turn through its corner, and subscript 1 is the state the body ahead delivers, 2 the state just behind the corner.

That is a curve that starts at p₂ and walks one way, toward p_c. It can only do that if the gradient just behind the corner points at p_c. The report keeps the form only for η ≥ 0 (p. 13) and says that at η = 0 “all equations reduce to those given by the generalized shock-expansion method”, whose pressure is simply constant along the element.

On a near-flat flare the two disagree, for a reason you can picture. The flow arrives at the flare along a long tube, where the pressure is still climbing back toward the free stream’s after the nose let it down: it is below the free stream and rising. The flare’s corner compresses it a little. Turn far enough and the pressure lands well above the tiny cone’s, and the gradient behind the corner turns downward with it, so everything agrees and the method runs. Turn less and the pressure stays below the cone’s, still rising toward it, so again everything agrees. In between there is a band where the compression has already carried the pressure just past its cone’s value while the tube’s own climb still pushes it further away. It has to rise, overshoot and come back, and one exponential cannot rise and fall. So η is negative there, and the element is reduced to the generalized method (issue #81: the method’s limit near a sharp tip is the same reading on a nose).

The region’s two edges are solved from the corner, not searched for. Each of the two quantities whose signs must agree is a smooth function of the flare’s turn, and on every corner state checked here each has a single zero, so the signs disagree on the open interval between those two zeros and nowhere else:

Single zero is not a promise, and a corner that breaks it exists: on a 25° cone with 20 mm of tube behind it at Mach 7, the pressure meets its tangent cone’s three times, at about 0.91°, 7.3° and 24°, so a flare there is reduced from 0.91° to 3.88° and again from 7.3° to 24°. flare_reduction_turns_rad sweeps the widening turns at a hundred stations before it brackets, so it reports that corner rather than handing back whichever of the three roots it walked to (a_corner_whose_gap_has_three_zeros_is_refused_rather_than_guessed_at). A pair of roots inside one station would still slip through, and the two turns would then be reported as one band when they are two.

Which of the two is the shallower is not fixed either: on this rocket the crossing is below the balance from Mach 1.5 up, and below that the order swaps.

Both are properties of the flow the body hands to the corner: its Mach number, its pressure, the gradient it carries, the radius there, the angle ahead and the free stream it was read in. flare_reduction_turns_rad takes exactly that and nothing else. The balance is TN 3527’s eq. 4 set to zero and rearranged, sin(δ₁ + θ) = (Ω₁/Ω₂(θ)) (sin δ₁ + r (∂p/∂s)₁ ⁄ B₁), which iterates on itself. The crossing is p₂(θ) = p_c(δ₁ + θ): an isentropic turn on one side, a cone solution on the other. It is bracketed over the turns a widening corner can make at all, from a surface lying along the axis up to the isentropic turn running out or the cone tables’ 30°, whichever comes first, and found by false position from the turn that would bring the pressure back to the free stream’s.

On the tests’ flared rocket (an ogive nose 0.25 m long on a 27 mm radius, a 0.7 m tube and a 0.3 m conical flare) they are these:

the_turns_a_reduced_element_lies_between_come_from_the_corners_own_state, in shock_expansion.rs, checks at eight Mach numbers that an angle a millionth either side of each edge falls on the right side of the march’s own reduction, and that the midpoint between them is reduced. That is a spot check at each edge, not an exhaustive sweep of the angles in between.

Those are the same numbers issue #117: the near-flat band the march used to refuse reported after bisecting the model’s own refusal (the band it quoted, 0.03816127° to 0.05882052°, is the crossing at Mach 4.70 and the balance at Mach 5, and the 0.00090182° it quoted is the crossing at Mach 2.20), but they are now read off two equations rather than found by trying the whole model against a sign test. Note what the table shows and that band hides: the region is not one interval in the angle. It moves with the Mach number, so a 0.04° flare is reduced at Mach 4.70 and marched at Mach 2.

How far to trust those digits. Not the search’s accuracy any more, but the tangent cone’s. Up to a half-thousandth of a radian (0.029°) hpr’s cone flow is slender-cone theory’s closed form, and the crossing closes to the last bits of an f64; the residual it leaves in the pressure is under 2e-14 at Mach 2.00, 2.20 and 3.00. Above that angle the cone flow is a Taylor–Maccoll integration, blended with the closed form up to 0.0573°, so the two edges past Mach 4 are read off the blend; there the residual is that integration’s own, about 1e-10 of the free stream’s pressure. Divided by how fast the gap closes with the turn, that is about 2e-10°, close to the 2.6e-10° the three operating systems CI runs were seen to spread the band’s lower edge over, so that spread was the cone’s and not the bisection’s.

Neither residual is promised to be zero, and on a body whose cone flow is harder it is larger: 4e-9 of the free stream’s pressure has been seen on a fatter body at Mach 2.6. So flare_reduction_turns_rad returns both of them beside the turns, and a caller who needs the digits should read them.

What hpr does now, and what changed. A reduced element is read by the generalized method wherever it has a tangent cone of its own: constant pressure and constant loading along it, which is what the report says the equations become. Before M1.8e19 that reading was allowed only on the nose, and anywhere behind it the march refused. Because hpr builds a body’s table of the method’s shares downward from Mach 5 and needs the whole 0.3 Mach of the join inside it, one refused row near the top took the table away and dropped the whole body to slender-body theory at every speed. Two switches came of that, and both are gone:

The sizes in the middle column are still measured, because they are the size of the fallback the model used to drop to: a_near_flat_flare_marches_every_row_and_the_fallback_is_still_measured, in model.rs, reads the same rocket on SupersonicFlare::SlenderBody and finds them again. The table’s start is now Mach 1.2 at fifteen flare angles from 0° to 1°, including 0.00024°, 0.00025° and 0.0003°: three angles a hair apart that used to give three different answers.

What is left is the crossing, and here is how big it is. At the crossing itself η has a pole, and that leaves a step, not in the pressure, which rides through because the gap it multiplies is zero there, but in the loading. On the side the method still owns, η runs to +∞ as the turn approaches the crossing, so the element sheds its corner’s loading onto its tangent cone’s within its own length; on the reduced side it holds the corner’s. The two differ, so the reading steps. On the whole rocket at 4°, measured either side of that Mach number’s own crossing with a ±1e-9° probe by a_near_flat_flare_reads_through_and_leaves_only_the_corners_crossing, in model.rs (Mach 4.95 rather than 5 because the body’s normal force stops at Mach 5):

A step, not a slope, except on the first row. Widening the probe a hundredfold, to ±1e-7°, leaves the figure where it is at Mach 3, 4 and 4.95, which is what says it is a step rather than the reading’s ordinary movement. At Mach 2 it does not: +0.00032% is already about what the reading itself moves over a ±1e-7° probe there, so that row is an upper bound on the step, not a measurement of one.

Those four numbers are one rocket, and they are not a bound. The step is the loading’s gap at the pole, so it grows with the length of the element that holds it, and where the region sits depends on the body ahead. Read on the body alone at Mach 5, either side of that body’s own crossing (what_the_crossing_costs_is_the_loading_gap_times_the_element_that_holds_it, in shock_expansion.rs):

Read the third and fourth rows twice: with a short shoulder the region is not near-flat at all: it sits between 0.7° and 4.6°, which is where real flares live.

But there is a bound, and it is exact: you can work out your own. At the crossing the two sides take the two constants the method relaxes between: the side it still owns sheds the corner’s loading onto its tangent cone’s at once, Λ_c = tan δ₂ (dC_N/dα)_tc, and the reduced side holds the corner’s, Λ₂ = (λ₂/λ₁) Λ₁. Along a conical flare both are constant, so eq. 19’s C_Nα = (2π/A_ref) ∫ Λ r dx integrates a constant and the whole step is

ΔC_Nα = (2π/A_ref) (Λ₂ − Λ_c) · ½(r_fore + r_aft) · L

for a flare of length L between those radii. flare_reduction_turns_rad returns Λ₂ − Λ_c beside the turns, as crossing_loading_gap_per_rad; on the tests’ rocket’s body at Mach 5 it is 6.116195e-4 per radian, and the formula reproduces the measured step to a part in 1e5 at flare lengths of 0.3, 1, 2 and 5 m. So the four rows above are an illustration of that formula, not the claim: the claim is the formula, and it holds for any conical flare.

The worst of the whole-rocket rows is a 64th of the switch it replaced in the force and a 227th of it in the center of pressure; the step at Mach 4.95 against the switch measured at Mach 3, so that is a comparison of sizes, not of the same flight condition, and the body-alone numbers above are not that small.

And the same pole is crossed in the Mach number. At a fixed flare angle the corner’s crossing sweeps past it as the speed changes, and the table’s rows are 0.05 Mach apart, so a flight reads it as a step between two neighbouring rows. On the short-shouldered body above with a 1° flare, the rows from Mach 2.90 to 2.95 step −2.77% and 0.14 calibres, where the neighbouring rows move by a fifth of that or less. Before M1.8e19: the near-flat flare those lower rows were refused, so the table began above them and the join covered the pole; it is now inside the table. That is the milestone’s trade, and it is the honest description of it: two switches in shape removed, one pole exposed in both shape and speed. On a body whose reduced rows sit at the bottom of its table, what is given up is a continuous join; what is bought is that an arbitrarily small change of shape no longer moves the whole body between two models.

It is not a new question, though: it is the loading through a tangent-cone crossing, which is what a crossing costs inside a segment, and which is open as issue #108: the loading through a crossing. This part of it, at Mach 2.90 to 2.95, is inside the operating envelope’s extended band, so it is part of M1.14h: the extended band; only #108’s part above Mach 4 is deferred (ADR-143). The region’s other edge, the balance, has no step at all: η is zero there, so the exponential form and the generalized method are the same reading, and the two branches meet.

What it leaves out.

• A cylinder’s and a boattail’s reduced elements are still refused, and still take the whole body off the method. Not because a flare relaxes toward something and they do not; a reduced element does not relax at all, on a flare exactly as on a cylinder: it holds the pressure and the loading behind its corner for its whole length. What differs is what it is read against. A widening element’s p_c and Λ_c are a real cone’s at the flow’s own Mach number, so the two readings either side of the region are two readings of one picture and they meet at the balance. A cylinder’s Λ_c is identically zero and its p_c is the free stream’s, so there is no cone there to meet, and a boattail’s is footnote 8’s constant (a stand-in, not a solution of that element’s own flow). Nothing here measures what that would be worth, so the refusal stands: issue #123: a cylinder’s or a boattail’s reduced element. How near is it? An ogive nose on a tube (0.25 m on a 27 mm radius, with tubes of 0.7 m, 3 m and 6 m) marches every row from Mach 1.2 to Mach 5 with nothing reduced, so it is not something a plain rocket walks into. What does hit it is TN D-4865’s own Newtonian start on the Arcas Robin’s nose from Mach 3.96, which is a reason hpr does not use that start (The two starts).
• The generalized method is the older, rougher one. Reading an element with it is a real choice, not a formality, and TN 3527 does not say it is what it would have done. What is checked here is that the choice joins the second-order reading continuously at the balance and that it makes the reading smooth in the flare’s angle, not that it is closer to a measured flare. No measured flare in this band exists.
• Below about a millionth of a radian the flare is not drawn at all. Corners turning by less than that are merged into the element ahead of them, because two tangents that nearly coincide meet at an ill-conditioned point. That covers every flare shallower than about 0.00006°, which at the bottom of the table’s range swallows both turns: at Mach 1.2 the crossing is 0.00000023° and the balance 0.00000018°, so there is no corner there to reduce and nothing the reading could switch on.
• Only the flare’s own element was measured. The two turns are solved for a corner behind a body; where a body has several corners that could be reduced at once, nothing here says how their readings combine. Nor does anything here say whether the generalized reading or the slender-body fallback it replaced was the nearer of the two to the air: what is claimed is that one of them moves smoothly with the shape and the other jumps.
• There is no rule of thumb for “is my flare one of these?”. The region belongs to the corner, so the only way to ask is to run the model: ShockExpansionBody::aft_flow on the body ahead of the flare, at the Mach number you care about, then flare_reduction_turns_rad on what it returns.

A step in radius

In short: a step is a joint where one part’s radius does not match the next one’s, so the rocket’s outline jumps rather than bending: a 54 mm tube butted straight onto a 75 mm one, or a coupler left standing proud of the airframe. That stops hpr’s second-order shock-expansion method, the march: it walks a chain of straight elements, the tangent body, from the nose tip aft, and it needs an outline without a jump in it. What hpr does about that is drop the whole body to slender-body theory, at every speed: a rocket with a step reads as though the method did not exist. M1.8e15 measured what that costs and tried the obvious fix; the fix was worse, so the behavior is unchanged and the cost is published instead. The decision record is ADR-049; the work left over is issue #87: a step takes the whole body off the method, open and not on the roadmap. Nothing measures a stepped body faster than sound, so neither the present reading nor any replacement has a reference.

What it costs. On the tests’ straight rocket (a tangent-ogive nose 0.25 m and three tubes of 0.7, 0.05 and 0.3 m, all 27 mm in radius) at Mach 3 and 4°, with the reference diameter pinned at 54 mm so that every row is divided by the same area. With no step it reads C_N = 0.899592, its center of pressure 16.9492 calibres aft of the nose tip. Each row is the change from that. Down means the body narrows from that joint aft, up that it widens:

At the threshold the shape is flush to a part in a billion either way, so the whole difference there is the method itself, the same wherever the step sits and whichever way it goes. Past that the shape itself starts to matter, and the two directions part: a step down takes area off the body and costs more, a step up adds area that carries slender-body normal force of its own and costs less. The center of pressure moves aft in every row, so a rocket that trips this reads more stable than the same rocket drawn flush. That extra margin is more likely optimistic than real: the model it falls back to reads 15% to 50% below the wind tunnel on the Arcas Robin’s body faster than sound (The body faster than sound in a flight). If you can draw the joint as a short transition instead of a butt joint, the body keeps the method; hpr’s own radius_step warning (the design model’s checks) is what tells you a design has tripped this.

a_step_takes_the_whole_body_off_the_method pins every figure in this section and the flush rocket’s own readings with it; Checking a claim says how to run a named test.

Where the threshold is. It is a pair, not one number, and which of the two binds depends on the joint:

Both are bisected. The second is 208× finer, and it depends on the body’s length and the change of slope rather than on its radius, so it is not a property of the step at all. It bites on the commonest high-power shape there is: on the tests’ finned rocket, a boattail whose fore radius is 27.0000000000002 mm rather than 27 mm loses the method for the whole body, worth −11.34% and 1.0951 calibres, larger than the tube-to-tube switch above.

Neither number is a judgement about steps. They are the widths of the rounding the tangent body can absorb, and every step anyone could build or draw is far past both, so in practice a step always takes the body off the method. (A third, much looser test, the run’s own coverage gate, which asks how much of the body the method can cover, at a millionth of the fore area or 13.5 nm of radius here, is what actually refuses every step bigger than that. It gives the same reading, and the test pins which of the two owns which range.)

What a fix has to handle. The obvious fix is to stop the march at the step and let the body ahead of it keep the method, the way the run already ends at a flare. That was built and measured, and it failed three ways. The numbers in this list were taken on that prototype, which was not kept: unlike the tables above, no committed test reproduces them, and ADR-049 records how each was measured and where the prototype lives.

• What is behind the step. ADR-034, the decision that the method covers a body or nothing, rejected mixing the two models on a measured case. A step’s remainder is supposed to be a plain tube, but “the run stopped at a step” does not make it one: with a boattail behind the step, the mixture’s center of pressure lands at 16.7209 calibres, forward of both pure models, the method’s 16.7286 and slender-body theory’s 17.8237. A reading outside the envelope of both models it is made of is the pathology ADR-034 measured.
• It does not close the band it was meant to close. The prototype removes the step’s switch at a tube-to-tube joint, but at a joint whose slope changes it is the corner ordering that refuses the body, at 1.3e−13 m, so a boattailed rocket still loses the method, worth −11.34% and 1.0951 calibres. The commonest shape it was supposed to help is the one it does not.
• Which shape stopped the run, not whether the joint was flush. The prototype keyed off the joint: any reason the run closed (a non-conical flare, a lip out of its wake) kept the forebody marched as soon as its fore radius was a picometer off, a new jump of +7.2% and 0.69 calibres where today the reading is continuous. That one is a property of how the prototype was built rather than of the idea, and a fix keyed off the shape would not have it; it is listed here because it is what a fix has to get right, not as evidence the idea cannot work.

What it leaves out.

• The step’s own force is slender-body theory’s, at any speed. (2/A_ref)ΔA at the joint, the limit of a transition whose length goes to zero; [B67] p. 18 assumes no discontinuities, so this goes beyond its source, and there is no compressibility term. A forward-facing step at supersonic speed stands a detached shock with a separated pocket ahead of it; none of that is modeled.
• No source gives a stepped body’s normal force faster than sound. MIL-HDBK-762 treats a rearward-facing step only as base drag, TN 3527 ([SD56]) needs a continuous profile, and nothing else pinned here covers one. So “take the whole body off the method” is not known to be right either; it is the reading that does not mix two models, which is the only argument for it.
• One body, one placement sweep. Three joints on one rocket at one Mach number and one angle, and the large steps up carry a moving reference diameter with them.

Blunt tips

What this covers: the body faster than sound when the nose’s tip is blunt or vertical, as on power-series noses with n below 1, Haack series (the von Kármán and L-V Haack) and elliptical noses, whose profile leaves the tip at 90°. How far to trust it: the cap comes from a NASA method checked only on spherical caps. On that report’s own sphere-cone, compared as its tunnel measured it (at the plotted angles, body lift included), hpr reads −1.2% to +32.1%: close through Mach 2.3, high from Mach 2.96, where the report’s own method reads +5.2% to +13.6%. On the Arcas Robin’s power-series nose the cap is an extrapolation. There, like for like, the body reads +37.2% at Mach 1.5 and +13.7% to +25.9% from Mach 1.8 to 2.96, and within 5% past Mach 3. Against the smooth secant ogive fitted to the same nose it reads lower at every Mach number: closer to the tunnel at nine of the eleven rows, by 0.7 to 5.8 points, and past Mach 4 it crosses into under-prediction and lands 0.6 to 1.5 points further out. A nose that is nearly a cone but for a vanishing tip carries a bias nothing here measures (issue #101, below). No validation flight reaches the speeds where any of this applies.

Why a cap. The shock-expansion method replaces the nose by straight elements, short cones and frustums each tangent to the profile, and starts at a pointed tip, where the air flows as it does over a cone. A vertical tip has no such cone: the shock stands off the nose, and the air just behind it is slower than sound. Jackson, Sawyer and Smith ([J68]) handled blunt noses by giving the tip’s cap Newtonian pressures and handing over to the method where the flow behind the cap is fast again, the handover. hpr does the same.

The cap. Newtonian theory takes the pressure from the angle δ between the surface and the wind: C_p = C_p,max sin²δ ([J68] eq. 1, p. 5). C_p,max is the pressure coefficient at the stagnation point, the tip, where the air comes to rest behind a normal shock: it follows from the pitot pressure a probe would read there, the Rayleigh pitot formula ([R1135] eq. 100). At a small angle of attack the windward side meets the wind a little more steeply, so the cap carries C_p,max sin δ cos δ of loading in the method’s terms (a hemisphere then carries its Newtonian drag turned into the body’s axes, C_p,max/2, as it must; test a_hemisphere_carries_its_drag_turned).

The handover. The method takes over where the surface’s slope falls to the largest angle a wedge can turn the flow through with its shock attached: 12.1° at Mach 1.5, 22.97° at Mach 2 ([R1135] eqs. 138 and 168). The report chose this point “simply because it gave the best agreement with the available data in the low supersonic-speed range” ([J68] p. 5). hpr caps the handover at 24°, which the wedge’s angle passes at Mach 2.06. The cap is there because the method needs the normal-force slope of a cone tangent to the body, and TN 3527’s chart stopped at 24° ([SD56] Fig. 2). Those slopes now reach 30° (ADR-042) and the cap has not followed, because the method’s march does not carry it that far. What the cap is worth measures what moving it would buy and what it would cost.

How much of the nose the cap covers depends strongly on speed. Where it ends, as a share of the nose’s length and of its base radius:

Near the join’s start a slender nose leans on Newtonian pressures over far more of itself than anything the report checked; by Mach 2 the cap is a percent or so of the nose, less than the report’s own. The join’s weight rises from 0 at its start to 1 a third of a Mach number later, which damps that, but read a vertical tip’s numbers between the join’s ends as the blend they are.

A power-series nose meets its base at a slope of n/(2f), with f its length over its diameter, and the cap can’t end on the nose while that is steeper than the handover’s angle. So such a nose takes the method no earlier than the Mach number where the handover’s angle passes its base’s, and its join to slender-body theory starts there rather than at Mach 1.2: for n = 0.5, Mach 1.23 at 3 diameters long, 1.49 at 1.2 diameters; the Arcas Robin’s from 1.22. One shorter than n/0.89 diameters (0.56 for n = 0.5) never takes it, since the handover stops at 24°, and keeps slender-body theory: hpr doesn’t warn, and AeroModel::supersonic_body returns None. Haack and elliptical noses end level, so the cap always ends on them.

A blunt nose can be more than one shape, and the cap may end on any of them. Since M1.8e18, a nose that starts with a sphere carries on through the curved, widening shapes behind that sphere, and hpr looks for the handover along all of them rather than in the sphere alone. TN D-4865’s own model 2 needs it: its nose is a sphere blended into a 2.75° cone by a second arc, and the sphere is still at 38.3° where the arc takes over, steeper than the 24° cap at any speed, so the handover always falls on the arc (What a marched flare is worth).

Everything else reads exactly as it did before, on purpose. A pointed nose is one shape however many curved shapes follow it, so a curved transition behind one is still the afterbody. And the search stops at the first shape that is straight or narrows: a cap that reached a cylinder or a boattail would hand the flow over at no angle at all, with none of the total pressure the tip took out of it, so a nose steeper than the handover all the way to one is still refused, which is the case in the paragraph above, and its numbers are unchanged.

Behind the handover. hpr starts the method there as it starts at a pointed tip, with the flow on the tangent cone, the cone that touches the body at the handover. The report starts it from the Newtonian pressure instead. What that choice is worth is set out in The two starts, after the checks below.

A worked example. The report’s sphere-cone at Mach 1.5: a nose radius of 0.175 base diameters on an 11.5° cone. The pitot pressure is 3.413 times the free stream’s, so C_p,max = 2.413/(γM²/2) = 2.413/(0.7 × 1.5²) = 1.532. The wedge’s largest angle is 12.11°, reached on the sphere 0.175 (1 − sin 12.11°) = 0.138 diameters behind the tip. On a sphere, with θ the angle from the tip, the loading C_p,max sin δ cos δ integrates over the cap to C_p,max sin⁴θ/2 on the sphere’s own cross-section; to θ = 90° − 12.11° that is 0.700, and 0.086 on the base (times 0.35²). The cone behind it, marched from the flow on a 12.11° cone, carries the other 1.595 of hpr’s 1.681.

How it was checked. Against the report’s own model 1, measured at Mach 1.50 to 4.63 ([J68] Fig. 8(a), p. 101, read from the scan by pixel analysis to about ±0.003, the plotting itself good to about ±0.01), compared as ADR-036 compares the Arcas Robin: hpr’s C_N at the plotted 0° to 12°, the method’s slope with body lift (Jorgensen’s, for a body of fineness 1.75, shorter than his Fig. 4 covers), fitted with a straight line just as the measured C_N and C_m are, and the report’s own method fitted the same way; per radian on the base, the center of pressure in base diameters from the tip:

Past Mach 2.3 hpr reads high twice over. At Mach 3.95 and 4.63 its slope at α → 0 is 13% to 21% above the measured one, and its body lift lifts its fitted slope 25% to 27% above that, where the measured curve rises only 9% to 16% above its own. The slopes at α → 0, beside it and not judged, with the measured one fitted with a curve two ways, as the decision record on comparing with a wind tunnel, ADR-036, asks:

And the Arcas Robin’s committed design, its power-series nose, cylinder and boattail with the lip left off, to show the cap’s own effect (the lip itself carries nothing: A lip in a boattail’s wake), through a flight’s path, fitted at the tunnel’s plotted angles as Checking the shock-expansion method fits them, beside the secant ogive fitted to the same nose:

The rows are in validation/fixtures/aero/blunt-tips.json, written by cargo xtask aero; the readings in tn-d-4865-sphere-cone.json. A test holds these tables to the fixture, cell by cell. The committed nose reads within 3.8 points of the fitted ogive up to Mach 2.96 and 5.1 to 8.2 points below it past Mach 3. Below it is not always closer: past Mach 4 its error changes sign, so at Mach 4.63 it reads −4.8% where the ogive reads +3.4%, 1.5 points further from the tunnel. Over the eleven rows it is nearer at nine. Below Mach 3 both read high, most at Mach 1.5, for the reasons in Checking the shock-expansion method.

What the cap is worth

A nose with a blunt or vertical tip flies Newtonian pressures over the tip and hands the rest of the body to the shock-expansion method where the surface’s slope falls far enough, the handover (Blunt tips above). That handover is never steeper than a cap, 24°. This section measures what moving the cap to 30° would buy and what it would cost. Nothing here changes what a rocket flies: the cap is where it was, and no committed number moved.

The cap could move because the cone slopes the method reads now reach 30° (ADR-042), where they once stopped at 24°. Moving it would follow the report’s own rule further: the wedge’s largest deflection passes 24° at Mach 2.06 and 30° at Mach 2.52, so a 30° cap keeps TN D-4865’s ([J68]’s) rule over that whole band, where 24° cuts it short from Mach 2.06 up. Where a cap binds at all, going all the way to 30° reads nearer the report’s own sphere-cone at every row, though not at every step of the way. And on the Arcas Robin’s committed nose it breaks the march (the method stepping element by element down the body from the handover) above Mach 4. hpr keeps 24° until that is settled (ADR-043), and the cap is a parameter of the method rather than a constant to argue over (with_handover_cap_rad).

Each cap starts to bind at its own speed (Mach 2.06, 2.19, 2.34 and 2.52), and below that it costs nothing at all. hpr’s error against TN D-4865’s sphere-cone, fitted as How it was checked fits it (hpr’s C_N at the tunnel’s plotted 0° to 12°, body lift included, fitted with a straight line), under four caps:

Below Mach 2.06 no cap binds, which is why the first two rows are one reading four times; at Mach 2.3 only 24° and 26° bind, so the last two columns agree. Read the first two binding rows with care for another reason: the cone slopes are tabulated from Mach 3 up and held at that row below it (Bodies faster than sound), so at Mach 2.3 and 2.96 a steeper cap’s gain is read off a slope that is not itself a function of Mach there. Where a cap does bind, the whole step from 24° to 30° reads nearer the tunnel by 0.4 to 1.7 points, most at Mach 3.95. The one place a steeper cap reads further out is the last step at Mach 4.63, where 28° reads +30.9% and 30° +31.3%.

Now the cost, which two counts tell you about. A reduced element is one where the method’s exponential law would run the wrong way: the pressure behind the corner heading away from the tangent cone’s instead of toward it, η < 0 in the method’s own terms ([SD56] p. 13), so hpr holds the pressure along it instead, issue #81’s open question. A crossing is the rarer and worse thing: the marched surface pressure passing through its own tangent cone’s, either way, within one part of the body. Where a nose meets a cylinder, a boattail or a flare the cone’s own pressure steps, which is not the same thing and is not counted. Both counts are per march, not per element. What a crossing is, and what it costs says how they differ and how far either one can be trusted (tangent_cone_crossings). Here are two of the four caps on the Arcas Robin’s committed power-series nose and the short model’s cylinder, nothing aft, C_Nα per radian on its cross-section at α → 0, read with the flown 10 elements per curve and with 160:

Through Mach 3.96 cutting the nose into sixteen times as many elements moves the answer by under 0.01 per radian under the flown cap and under 0.013 under the 30° one: the answer is the model’s, not the mesh’s. Above it the 30° cap’s march crosses its tangent cone twice, and reduces most of the nose along with it (5 of 10 elements at Mach 4.63 and 9 of 10 at Mach 5, which is what hpr would fly, and 109 and 145 of 160), and the answer follows the element count instead, and not even in order: 3.047 at 10 elements, 2.928 at the 40 the fixture also holds, and 3.260 at 160, a spread of 0.33 per radian, 11%, where the flown cap moves by 0.1%. The crossing is the cause and the reductions travel with it, which the next section takes apart.

What a crossing is, and what it costs

Throughout this section, p₂ is the surface pressure just behind a corner and p_c its tangent cone’s, Λ the lift per unit length (the loading) and Λ_c the tangent cone’s, all as the method’s equations define them.

The steeper cap starts the march from a steeper cone at a higher pressure, and from there the tangent cone’s own pressure falls away faster than the marched pressure does as the nose flattens. So the surface pressure catches its tangent cone’s and passes through it (at Mach 4.63 under the 30° cap, about a tenth of the way back) and stays above it until the nose flattens enough for the cone to catch up again. That is two crossings: one out, one back.

Why that hurts has nothing to do with η < 0. Write the exponent in e^(−η) as η = k (x − x₂), so that k = (∂p/∂s)₂ / ((p_c − p₂) cos δ₂) is a relaxation rate per meter: the gradient just behind the corner divided by how far the pressure has to go. A crossing closes that gap while the gradient carries on, so k has a pole: it runs to infinity. The pressure itself doesn’t mind, because k (p_c − p) cos δ₂ is only the gradient again, and that stays finite. The loading does mind, because it relaxes toward Λ_c at the same k ([SD56] eq. 19) while its own gap is set by something else entirely.

Follow that gap through the Mach 4.63 march under the 30° cap. Between the two crossings nearly every element is reduced, so its loading is held where it was and never relaxes: by the second crossing Λ stands about a quarter above Λ_c. The element at that second crossing is back inside the method, and how much of that quarter it sheds in its own length is whatever 1 − e^(−η) happens to be for the step the mesh gave it. On the 40-element march it sheds 98% of the gap in one step; on the 160-element march, 12%. That is the answer moving with the mesh, in one number (test a_crossing_is_a_pole_in_the_rate_the_march_relaxes_at).

The counts say the same thing over the whole sweep. Of its thirty-two readings (four caps at eight Mach numbers on this one nose), twenty-seven never cross and five do:

No overlap, and nearly three times (2.9×) between the two groups. The five are 30° and 28° at Mach 4.63 and 5, and 26° at Mach 5 (fixture blunt-tips.json, test over_the_sweeps_meshes_a_crossing_separates_the_readings_that_move).

A crossing is a flag, not a verdict, and it has to be read carefully. Three limits, all measured:

• A crossing does not prove an answer never settles. 28° at Mach 5 crosses at every mesh, and its 0.69 spread is all in the coarse end: from 60 elements to 640 it holds to 0.005 per radian, tighter than the worst reading in the sweep that never crosses. Compare 30° at Mach 4.63, which moves by more than 0.2 per radian over that same range. What the sweep shows is that across its three meshes (the flown ten elements per curve and refinements to forty and a hundred and sixty) the crossings and only the crossings mark the readings that move (test a_crossing_says_the_answer_moved_not_that_it_never_settles).
• A count of zero does not prove one settled. Whether a crossing is seen depends on the mesh: 26° at Mach 5 and 28° at Mach 4.63 show none at the flown ten elements per curve and two at forty and a hundred and sixty, and both move. Read zero as “not proven”.
• hpr does not count crossings while it flies, and everything here is one nose. The count is a tool for studying a body, not a guard, and another blunt nose above Mach 4 could cross under the flown 24° cap without saying so.

Reduced elements, on their own, do not move an answer. TN 3527’s own fineness-3 ogive reduces 27 of 160 elements at Mach 5.05 and 50 of 160 at Mach 6.28, and its answer settles to 0.002 per radian from 10 elements to 160. There η < 0 comes from the gradient changing sign, with the surface pressure below its tangent cone’s the whole way down; the gap never closes, so there is no pole. That ogive never crosses at either Mach number we can check it against the report at, which is why the report could state its condition ([SD56] p. 13) and stop: it never had to say what a crossing does (test a_reduced_element_settles_where_tn3527s_own_bodies_never_cross).

The two questions are tangled, though, and that is the state of play. The quarter-wide loading gap the second crossing sheds was opened by the reduced stretch behind it, which is hpr’s η = 0 reading, not the report’s rule. So a different reading of η < 0 would change the size of the step as well, and neither question can be judged without the other. What has changed is that the step itself is taken by an element the method still owns, so a rule for η < 0 alone is not obviously enough.

The disorder under the steeper cap is not rounding: nudge the Mach number by eight units in its last place (about a part in 10^15) and the same elements reduce, for an answer that follows to a part in a billion (test a_steeper_handover_moves_the_march_out_of_its_range).

The break is not at 30°, and it is not orderly. It sits between the flown cap and the next step, and 28° is the worst of the four: at Mach 5 its answer moves 0.69 per radian over the element count, twelve times the 30° cap’s 0.055. No cap above the flown one holds its answer to Mach 5:

Below Mach 4 the four agree to 0.013 per radian, so nothing here says a cap between the two ends is a middle ground. It says the flown cap is the last one whose answer is the model’s all the way to Mach 5.

Settled is not the same as right. Under the flown cap hpr still reads +29.7% and +32.1% against the sphere-cone at Mach 3.95 and 4.63, as the first table says. The cap chooses between an answer that is high and one that is high and moves with the mesh.

So the cap hpr flies is set by the march’s range rather than by a chart’s edge. It moves when the method has a rule for what the loading does where the surface pressure crosses its tangent cone’s: a rule whose answer stops changing as the nose is cut finer, judged together with the reading of η < 0 that sets the gap it sheds. TN 3527 does not state either, because its own bodies never cross, so this is a modeling decision rather than a measurement to look up (issue #108: a steeper handover crosses the tangent cone above Mach 4). The milestone that would then move the cap, M1.8e16, the blunt tip’s handover past 24°, waits on that (ADR-044, which records the measurement behind this section). Both are deferred as beyond the operating envelope, since the crossing bites above Mach 4; the vertical-tip switch’s error below that is part of M1.14d: supersonic accuracy, and of M1.14h: the extended band above Mach 2.5 (ADR-143). All three tables are held to blunt-tips.json by a test, cell by cell; the second shows the two ends of the sweep, and the fixture holds 26° and 28° and a 40-element reading too. Its numbers are stored to six decimals, three more than the tables quote: where most of the nose is reduced the march is not reproducible past about 1e-11 from one machine’s maths library to another’s, so pinning more would only break the build (ADR-043).

The two starts

hpr starts the march from the tangent cone at the handover, the report from the Newtonian pressure there. Read at a small angle, the report’s start fails on the Arcas Robin’s nose from Mach 3.96, where the march reduces the element at the nose’s end (holding its pressure where the method’s exponential law would run the wrong way) on the cylinder behind it, whose tangent cone is the free stream rather than a cone of its own, and which hpr therefore refuses (issue #123: a cylinder’s or a boattail’s reduced element); from Mach 2.96 its answer drifts as the nose is cut into more elements, for the same reason (issue #81, the method’s open question there). A flight’s table is built from Mach 5 down, so that failure would leave such a rocket no method at all. The tangent cone’s start holds to Mach 5 and settles: the Arcas nose moves under 0.01 per radian from 10 elements to 40, a five-calibre elliptical or von Kármán nose under 0.02 from 10 to 160. On the report’s sphere-cone, against the measured slope at α → 0, the report’s start reads closer than hpr’s at Mach 1.9, 3.95 and 4.63, about the same at 2.96 and further at 2.3, and 95% high at Mach 1.5. That last is hpr’s reading of the report’s start at α → 0, not the report’s method, which reads 1.844 there at its own angles: the handover sits 0.6° above the cone, so its linear range is that small. hpr takes the start that holds and settles everywhere over one that fits one body better where it holds. Both are kept: HandoverStart selects the report’s for comparison (ADR-038). Slopes per radian on the body’s cross-section, at α → 0, for the committed nose and the short model’s cylinder, nothing aft:

What it means for a rocket. Mostly more force, barely any change of balance. On Calisto, whose von Kármán nose now takes the method past Mach 1.2, the whole rocket’s normal-force slope rises 17% to 31% from Mach 1.5 to 2 (Normal force through Mach 1), while its center of pressure moves by under 0.15 calibres (forward at Mach 1.5, aft at Mach 2). So the stability margin moves by under a sixth of a calibre, and the force that holds the rocket into the wind grows by about a quarter.

What it leaves out.

• No measurement checks a tip that isn’t spherical. The report tested spherical caps only. Newtonian theory on the cap and the tangent cone’s start are both approximations, and the method’s reduced elements (issue #81) still apply behind them.
• A cap that shrinks to nothing doesn’t reach the cone it sits on (issue #101). The march carries its start cone’s total pressure the whole way, as the method does from any vertex, and nothing makes that fade as the cap shrinks. A power-series nose of n = 0.99 is a 7.1° cone but for a tip 1e-55 calibres across, yet at Mach 4 its cylinder carries 1.21 per radian where the cone’s carries 1.37, 12% less, because the march runs on the 24° cone’s total pressure rather than the 7.1° cone’s. The shapes a rocket really uses have caps that are small but not vanishing (at Mach 1.5 the table above puts their ends at 0.12 to 0.76 of the base radius, where that nose’s is 1e-55), and how much of this bias they carry is unknown.
• At α → 0 the handover is held where it sits on the body, as TN 3527 holds every other point. The report’s equivalent bodies turn the body about the sphere’s center, which slides the handover along the surface instead; hpr leaves that term out. How much it is worth is not measured here. The two starts in the tables above differ by more than it alone, since their pressure and total pressure differ too: 2.53 against 2.87 per radian on the Arcas nose at Mach 1.5, and 1.70 against 1.70 on the sphere-cone at Mach 2.96.
• Two switches in shape, of the family issue #87 tracks: a vertical-tip nose steeper than the cap’s handover all the way to its base gets no method at all, and a pointed tip steeper than the cone tables’ 30° is refused where a vertical one flies. The pointed tip’s edge was Fig. 2’s 24° until M1.8e11. The vertical tip’s edge is the handover’s cap, which stands at 24° for the reason above. Moving the cap waits on issue #108: a steeper handover puts the march into η < 0 above Mach 4, which is deferred; the switch’s error itself is part of M1.14d: supersonic accuracy (and of M1.14h above Mach 2.5).
• Elements that merge, merge with Mach. Behind the cap, a tangency point whose tangent turns by under a microradian is folded into the element before it, because its corner can’t be placed in floating point. Which points merge changes with the handover, so the method’s answer takes a step of about a millionth of a per-radian slope as it does: far below anything measured here, but there.
• Drag is unchanged: the nose’s wave drag already covers blunt shapes (Drag through Mach 1).

Fins

A fin set is N identical fins spaced evenly around a body tube. For one fin of the set:

• Trapezoids. tan Γ_c = (x_t + c_t/2 − c_r/2)/s, and the closed forms give [B66] eq. 57 and 76a exactly.
• Ellipses on the root chord. Γ_c = 0, c̄ = 8c_r/(3π), y_MAC = 4s/(3π) and X_f = (½ − 2/(3π)) c_r = 0.28779 c_r. Loft replaced the ellipse with an equal-area trapezoid, whose sweep made the slope 1.3% low (Loft lesson L10).
• Freeform outlines. c(y) runs from the leading edge to the trailing edge, so a jagged edge’s gap counts toward the CP but not toward A_fin ([N09] pp. 27–28). Γ_c is the span average of the mid-chord angle ([N09] p. 29), which gives the natural angle for trapezoids and ellipses. The integrals are exact: between vertex heights the edges are straight, and a three-point Gauss rule (quadrature, a weighted sum of samples) per band is exact. Bands thinner than 1e-12 of the span (vertex heights a few rounding steps apart, as when a tip is converted from inches) are skipped.
• A root along a nose cone or a transition (ADR-166, fins on a nose cone). There the root (where the fin meets the body) follows the curved surface. Its points are part of the fin’s outline, so A_fin, c̄ and the CP are those of the region between the outline and the surface, with heights y from the surface at the root’s leading edge. The body radius r_t in the interference factor K_T(B) above is the radius there. No measured fin on a nose cone checks the model; against OpenRocket, the one example is the cockpit of Pods–airframes and winglets:
  • Across the airflow at Mach 0.3, its normal-force slope is 0.2784 per radian in hpr and 0.2117 in OpenRocket, 31.5% more, with the CP 0.13 mm apart. Closed straight along its chord, hpr’s would be 0.2802, so the curved root isn’t the cause (#326).
  • With the air crossing at 0° roll, the direction OpenRocket’s margin is taken in, the single fin lies in the airflow’s plane, and neither code counts it there. hpr’s margin is the weakest direction’s since #329 (Stability margin).
  • Above Mach 1, the tip-cone correction still mirrors the flow at a level line through the root’s leading edge, an approximation on a root that rises.
• Prandtl–Glauert enters through β in the fin slope only, up to Mach 0.8. The CP stays at the quarter chord, a quarter of the way along the MAC, through that range ([B67] p. 6). Past Mach 0.8 the fins follow Fins through Mach 1, below.
• Fin count. A fin at angle Λ_k to the lateral airflow adds (C_Nα)₁ sin² Λ_k in the plane of the flow. The sum is N/2 for three or more evenly spaced fins, at any roll. For one or two fins it changes with the direction the air crosses, so such a rocket has a different margin in each direction; hpr reports the weakest (Stability margin). f_N is 1 up to four fins, then 0.948, 0.913, 0.854 and 0.810 for five to eight ([TD] eq. 3.54). Those factors make six and eight fins 1.37 and 1.62 times four ([762] p. 5-24), and interpolate five and seven (Loft lesson L8). More than eight fins are refused: [TD]’s 0.750 has no data behind it. [N09]’s roll-dependent 15% and 6% reductions for three and four fins were dropped in [TD].
• Side force of one- and two-fin sets. Each fin sees α sin Λ_k ([N09] eq. 3.50) and pushes along its own normal. Eq. 3.51 keeps the in-plane share sin² Λ_k; the share across the plane is sin Λ_k cos Λ_k, which cancels for three or more fins but not for one or two. [N09] pp. 31–32 drops it, arguing that it cancels for two or more fins; for two fins the pushes add. hpr reports it as C_Y at the fins’ CP (derived here from eq. 3.50, not taken from a source).
• Interference K_T(B) is Barrowman’s straight-line fit to NACA TR-1307, justified for r_t/(s + r_t) < 0.4 ([B66] p. 36).
• Not modeled.
  • The body lift the fins induce, K_B(T) ([B66] p. 36 neglects it; [B67] eq. 3-98 has it).
  • The roll moment of a single fin’s normal force at an angle of attack: its force acts at r_t + y_MAC along the fin’s normal. Two or more even fins cancel it; one fin doesn’t. Roll from cant and against the roll rate is modeled (Roll: forcing and damping).
  • Interference between fin sets at the same station.
  • Damping coefficients for pitch and yaw. They would replace the local-flow damping below, not add to it, or it would be counted twice; hpr keeps the local flow:
    • In a flight, pitch and yaw damping come only from evaluating each component in its own local flow, which includes the speed the rocket’s rotation adds there (Rigid-body flight).
    • Only components with a slope give it: nose cones, transitions and fin sets. A boattail’s slope is negative, so it takes some away.
    • Body tubes give none at small angles. Their own slope is 0, and their body lift grows with sin² α, so it adds nothing there.
  • Tube fins, which have their own section (Tube fins). Any part kind the model doesn’t know is refused. Pods have their own section (Pods).
  • Launch lugs and rail buttons add drag only.

Fins through Mach 1

Faster than sound, air can’t flow around a fin’s edges ahead of it. The fin’s lift comes from the pressure behind the shock and expansion waves at its surfaces, and a different theory applies. hpr uses Barrowman’s subsonic method to Mach 0.8, supersonic linear theory from where that theory holds, and a straight-line join between them. The body terms don’t change with Mach: slender-body theory’s slope and CP hold at any speed ([B67] p. 18). How well this agrees with a wind tunnel and with RASAero II is under Verification.

Supersonic linear theory. Past Mach 1, β is redefined as √(M² − 1), which grows from 0 as the speed passes that of sound. A thin flat plate at a small angle α to a supersonic flow has the pressure coefficient +2α/β on the side facing the flow and −2α/β on the other: the first term of the pressure series Barrowman uses ([B67] appendix A, p. 82). So every part of the fin carries the same load, 4α/β per unit area, and each strip (a narrow slice of the fin along the airflow) carries it at its middle. MIL-HDBK-762 finds linearized theory accurate for the supersonic stability of thin fins ([762] p. 5-15). Two things change the load near the tip:

• The tip’s Mach cone. The tip disturbs the flow only inside the cone that spreads inboard from its leading edge at the Mach angle, atan(1/β). Barrowman halves the load inside it ([B67] appendix A, p. 84). For a rectangular tip that is exactly linear theory’s loss.
• The body as a mirror. At the root the body stands in for the fin’s mirror image. A cone that reaches the root continues into the mirror image, and the part of it there counts too, as the mirror fin’s cone crossing onto this one.

A_cone is the part of the fin (and of its mirror image) inside the tip’s Mach cone. The fin-count factor, the sum over fins and K_T(B) apply as above.

Where it starts. Linear theory’s strips need four things, so it starts at M_s = max(1.2, 1/cos Γ_L, 1/cos Γ_T, √(1 + 1/AR²), √(1 + (c_t/2s)²)), where Γ_L and Γ_T are the leading- and trailing-edge sweeps, AR = 2s²/A_fin is the aspect ratio of the fin and its mirror image, and c_t the tip chord:

• Mach 1.2, the bottom of the supersonic region ([N09] Table 3.1, p. 19);
• supersonic edges, each with its Mach number square to the edge, M cos Γ, past 1, the case [TN2114] covers;
• β·AR ≥ 1, where linear theory’s tip loss holds; a rectangle’s slope peaks there, at 2·AR;
• β ≥ c_t/(2s), so the mirror fin’s tip cone stays off this fin’s tip, which matters for a tip chord longer than the fin’s average.

Calisto’s and the Arcas Robin’s fins start at their leading edge’s 1.2806 and at 1.2.

The transonic join. From Mach 0.8 to M_s, the slope and the CP are each a straight line in M between their values at the two ends. No source gives this region in closed form. MIL-HDBK-762 reads it from charts of transonic similarity (the way thickness and Mach number combine near Mach 1, [762] pp. 5-104–5-105). The join keeps both continuous. For most fins the slope peaks at M_s (a leading edge swept forward can make it fall across the join instead), and the CP moves aft from the quarter chord toward the middle of the chord.

Worked example. Calisto’s 2018 fins: root chord 0.12 m, tip chord 0.04 m, span 0.10 m and sweep length 0.08 m, on A_ref = 0.012668 m² (d_ref = 0.127 m). The leading edge is swept 38.66°, so M_s = 1/cos 38.66° = 1.2806. At Mach 2, β = 1.732. The tip cone is a triangle 0.04 m along the tip and 0.04/β = 0.0231 m down the unswept trailing edge: 0.000462 m², 5.8% of the fin’s 0.008 m². So (C_Nα)₁ = (4/1.732)(0.008 − 0.000231)/0.012668 = 1.416 per rad. hpr gives, per fin:

What it leaves out.

• Thickness. Linear theory is for thin plates; a thick fin or a blunt leading edge detaches the bow shock near Mach 1.
• Exact linear theory for a tapered fin. The strip method’s constant load outside the tip cone runs above it. Against [TN2114] eq. A7 (printed p. 18), a fin with a taper ratio of 0.5, an unswept trailing edge and βA = 3 gets 4.5% more slope.
• Subsonic leading and trailing edges, which the join covers without a method of its own. A curved edge counts by its span-averaged sweep, so an elliptical fin, whose edge is swept 90° at the tip, or a freeform fin with a raked outboard edge, keeps a subsonic stretch past M_s.
• Leading edges swept forward. The tip then sits ahead of the root and its Mach cone covers much of the fin, where the half-load overstates the loss, so the slope rises past M_s instead of falling: by up to +7.7% for fins swept 40° to 50° forward, peaking up to 0.37 Mach later (issue #64). Such fins are rare on rockets. A property test holds every trapezoid whose leading edge is straight or swept aft (to 65°), tapered either way, to a slope that falls with Mach from M_s.
• The fins’ lift carried onto the body behind them, K_B(T), as below Mach 1.

Other choices, and why not.

• Niskanen’s supersonic slope ([N09] eq. 3.48–3.49) multiplies the fin’s area by one strip’s pressure coefficient, K₁α + … with K₁ = 2/β: the pressure on one face. A plate is pushed by the difference between its faces, twice that. His thesis finds its simulated C_Nα for the Arcas Robin “notably lower than the experimental values”, with the cause unknown (p. 91, a comparison that runs to Mach 4). hpr counts both faces.
• RocketPy 1.13.0 flies Diederich’s subsonic slope at every Mach number, with β held at 0.6 from Mach 0.8 to 1.1; past Mach 1 that tends to 2π cos Γ_c/β, about π/2 times linear theory’s 4/β. Its fin CP doesn’t move with Mach.
• Tuning the join to the wind tunnel below would shrink its misses by fitting the model to its own check. The join’s ends come from the sources’ speed regions, set before measuring.

Roll: forcing and damping

Fins set at a small angle to the rocket’s axis, cant, each push sideways as a wing at that angle would. Each push acts off the axis, so together they twist the rocket and spin it up: the roll forcing. Once the rocket rolls, each fin also moves sideways through the air, meets it at an angle of its own, and pushes back against the spin: the roll damping. The two balance at a steady roll rate that grows with the airspeed. hpr takes both from Barrowman’s thesis ([B67] §3.13–3.14, appendix A), by strip theory: each narrow strip of a fin, running with the flow, lifts in proportion to the angle it meets the air at.

How far to trust it (Roll against the Arcas Robin and the Basic Finner):

• The forcing is within 5.3% of NASA’s measured roll effectiveness (the rolling moment per degree of cant) from Mach 2.3 to 4.63, and reads 14% to 48% high at Mach 1.5 and 1.8.
• The damping reads 6% to 16% low against the one measured set, from Mach 1.5 to 3.
• Below Mach 1.5, where most hobby flights stay, nothing measured checks either: only the flight’s agreement with the closed-form balance, and Barrowman’s own computed damping at Mach 0.07. Strips that each lift at the fin’s average slope ignore how the flow at one strip changes the next, which for short fins likely overstates the damping, so a subsonic spin may read low.
• The steady spin rate carries both errors, and both push it high: forcing that reads high and damping that reads low each raise it.

Other symbols are as in Fins: r_t the body’s radius at the fins, s the span, c_r and c_t the root and tip chords, A_fin one fin’s area, y_MAC the mean aerodynamic chord’s distance from the root.

The moment. A fin set of N fins adds C_l = −N C_lδ k_T(B) δ + N C_lp k_R(B) (p d/2V). The minus sign is the cant’s direction: a positive cant turns fin 0’s leading edge toward −y_B, so its lift pushes toward −y_B and turns the rocket about −z_B (a negative C_l), clockwise seen from ahead of the nose, looking aft. Its first term, summed over the sets, is C_l0. In a flight the cant’s forcing is scaled by cos α: the cant meets the air as it runs along the axis, so there is none broadside and it reverses tail first, as the fins’ normal force follows sin α (Rigid-body flight). Bodies of revolution add nothing, and fin–fin interference is left out, as in [N09] eq. 3.66.

Below Mach 0.8. The cant is an angle of attack for each fin, so its lift is the fin’s own normal force, acting at its mean aerodynamic chord: C_lδ = (C_Nα)₁ (r_t + y_MAC)/d ([B67] eq. 3-35, [N09] eq. 3.66). For the damping, a strip at ξ meets the air at −pξ/V, and lifts by the fin’s slope per unit of its area, a = (C_Nα)₁ A_ref/A_fin: C_lp = −2a ∫ξ² dA/(A_ref d²) ([B67] eq. 3-40–3-49, [N09] eq. 3.67–3.70). For a trapezoid ∫ξ² dA = (c_r + c_t) r_t² s/2 + (c_r + 2c_t) r_t s²/3 + (c_r + 3c_t) s³/12; for any outline hpr takes it from the polygon.

From M_s, where supersonic linear theory starts (Fins through Mach 1), each strip carries the load 4α/β, halved inside the tip’s Mach cone: C_lδ = (4/β)(∫ξ dA − ½∫_cone ξ dA)/(A_ref d) and C_lp = −(8/β)(∫ξ² dA − ½∫_cone ξ² dA)/(A_ref d²), with β = √(M² − 1) ([B67] appendix A, first order). Between Mach 0.8 and M_s each is a straight line in M, as the fin’s slope is.

The body. The body reshapes the flow the fins meet. Barrowman’s factors from slender-body theory ([B67] eq. 3-95 with 3-105, and 3-123 with 3-122), with τ = (s + r_t)/r_t, scale the forcing by k_T(B), 0.940 at τ = 2, and the damping by k_R(B), 1.33 at τ = 2 for a rectangular fin; both are 1 without a body. [N09] leaves both out. k_R(B) is for a chord that falls linearly from root to tip; another outline takes it at its tip-to-root chord ratio, so an elliptical fin is taken as a triangle, about 5.5% too much damping.

The steady roll rate is where the two cancel: p = −(C_l0/C_lp)(2V/d). Below Mach 0.8 the fin’s slope cancels between them, and for one fin set p = −δ V A_fin (r_t + y_MAC) k_T(B) / (k_R(B) ∫ξ² dA): it grows with the airspeed and the cant, and not with the air’s density or the number of fins.

A worked example: Valetudo with its fins canted 1°. Valetudo, one of RocketPy’s example rockets, has three fins 58 mm long at the root, 18 mm at the tip and 77 mm in span on a body of radius 40.45 mm. One fin has A_fin = 2926 mm², y_MAC = 31.75 mm, ∫ξ² dA = 1.656 × 10⁻⁵ m⁴, and τ = 2.904, so k_T(B) = 0.935 and k_R(B) = 1.228. At 100 m/s, p = −0.01745 × 100 × 0.002926 × 0.0722 × 0.935/(1.228 × 1.656 × 10⁻⁵) = −16.95 rad/s, 2.7 turns a second. With no drag and no gravity hpr’s flight settles on it within 1e-6 (1e-11 measured), spinning up with a time constant of 0.48 s (canted_fins_spin_to_the_analytic_balance, which also checks this example’s rate and time constant).

Why these choices:

• The fin’s own slope in the damping. Barrowman’s text writes the airfoil’s slope C_Nα0 there (eq. 3-40, 2π/β), as does [N09] eq. 3.69. His own computed curve for the Basic Finner reads −34.21 at Mach 0.07 (Fig. 5-7), which is the fin’s slope spread over its strips, −33.53 (the_basic_finner_damps_as_barrowman_computed); the airfoil’s gives about −81, 2.4 times as hard. His curve’s rise toward Mach 1, about 20% read from the figure, follows the fin’s slope too, where the airfoil’s would grow without bound. Stubby fins lift far less than an airfoil.
• The body factors. Barrowman has them and [N09] doesn’t. For the Arcas Robin’s fins they lower the forcing 6.5% and raise the damping 20%. His k_R(B) is a ratio of forces (eq. 3-116, 3-120) applied to a moment (eq. 3-123); weighted by the moment it would be 3.4% to 4.2% smaller for the fins here. hpr keeps his, which his computed curve seems to use too. k_T(B) is reference 23’s factor for fins turned together; for cant, whose load turns the other way on the opposite fin, it isn’t derived.
• Moments about the body’s axis. Faster than sound Barrowman’s appendix A takes each strip’s moment about the fin’s root; hpr takes it about the axis, ξ = r_t + y, as his subsonic eq. 3-27 and 3-35 do. About the root the Arcas Robin’s forcing would be about half: its load sits 25 mm from the root and 53 mm from the axis.
• Limits on the input. hpr refuses a cant beyond 15°, where a fin stalls and the linear model means nothing, and a cant on a single fin, whose sideways push it doesn’t carry.
• Pitch and yaw keep the local-flow damping. A flight’s pitch and yaw damping come from each part’s own local flow (ADR-011); coefficients would count it twice.
• No target was set for the comparisons. The roadmap asked for the forcing to be compared with the measured roll effectiveness, and set no bar; the numbers are reported as they are (ADR-031).

What it leaves out:

• The roll forcing near Mach 1.5 reads high: linear theory’s load rises as 1/β toward Mach 1, and the Arcas Robin’s measured forcing doesn’t. Barrowman found the same for the Tomahawk sounding rocket (“the theoretical value at M = 1.5 is no good”, [B67] p. 66).
• The damping reads low for the Basic Finner’s thick wedge fins, 8% of the diameter thick: first-order theory has no term for thickness, the likely cause.
• Nothing measured checks roll below Mach 1.5.
• A fast spin at low airspeed meets the fins at angles past stall, where the linear damping no longer holds: Valetudo spinning at 17 rad/s at 5 m/s meets the air 23° off at its fin tips.
• The angle of attack: the measured roll effectiveness changes by up to 13% between 0° and ±4° (TN D-4014 Fig. 14); hpr’s is the same at every angle. A single fin’s roll from its normal force, the body’s own roll, and fins’ airfoil sections are not modeled.

Pods

A pod is a body beside the airframe: a side pod, or an outboard motor pod (the design page’s Pods). hpr gives each pod the forces its parts would have on the airframe, once per pod, and adds them to the airframe’s. No measured flight checks it. The formulas are checked against hand-worked numbers. They are also checked against OpenRocket 24.12 on six probe designs, small designs written only to test pods: five carry pods of bodies, fins, a tail cone, winglets or motors, and one is the same airframe without pods (M1.13c2, a pod design against OpenRocket). Below Mach 0.81, hpr’s apogee is within 0.81% of OpenRocket’s, and its stability margin within 0.0014 calibres. What the pods change agrees to 0.32 percentage points of apogee (table below). That shows the two codes compute the same thing, not that either is right: both leave out how the pods and the body disturb each other’s flow, whose size is given below. Faster than sound a pod keeps slender-body theory’s slope, which nothing checks.

The rule. A pod’s nose cones, transitions and tubes are Barrowman’s bodies, as on the airframe ([B67] eq. 3-65, [N09] eq. 3.19): a part whose cross-section grows from A_fore to A_aft has the slope C_Nα = 2 (A_aft − A_fore)/A_ref at the center of pressure Bodies of revolution gives it, on the rocket’s reference area A_ref. Each pod adds that once:

• Normal force. N pods add N times one pod’s slope and moment, at the part’s station along the rocket. The pod’s first part steps up from nothing, as the airframe’s nose does, so a pod that starts with a tube gets no slope from its flat front. Faster than sound, a pod keeps slender-body theory’s slope, since the shock-expansion method (Bodies faster than sound) covers the airframe alone.
• Body lift. Each pod’s body lift, Jorgensen’s crossflow term, is taken on the pod’s own planform and at the pod’s own fineness (its length over its largest diameter), not the airframe’s.
• Fins on a pod. A pod’s fin set is turned with its pod: pod k at roll angle φ_k holds fin j at θ_j + φ_k, and each fin takes its share of the flow as an airframe fin does (Fins). The fin–fin factor (how a set’s fins shade each other) counts one pod’s fins. The body’s interference with the fins, K_T(B), is the pod tube’s: 1 for the tube of no radius a pod of winglets hangs from.
• Drag. Each pod adds its own drag buildup (Drag): friction on its surface, with the body form factor at its own fineness; its nose’s or shoulder’s pressure drag; its steps; its boattails; and its own base, the last part’s aft area. A pod’s fins, lugs and buttons drag once per pod too. The Reynolds number stays the rocket’s, as for every part ([N09] §3.4).
• Motors in pods. A burning motor’s cross-section comes off the base of the body it sits in: the airframe’s motors off the airframe’s base, and a pod’s off its own pod’s ([N09] pp. 50–51 for the base). This is worked out per pod set: each pod set that holds motor mounts has its own total of thrusting motor area (DragConditions::thrusting_pod_motor_areas_m2, in the order of Layout::motor_pod_sets), and each of its pods takes an even share of that total. A base smaller than its motors has no base drag left, never less. Up to MOTOR_POD_SETS, 4, pod sets may hold motor mounts; a rocket with more is refused by name. Until M4.5i, motor mounts in two different pod sets were refused (ADR-168). Worked example: two pod sets of one pod each, both holding a burning 18 mm motor (2.54 cm² each), take 2.54 cm² off each pod’s own base; the airframe’s base keeps its own motors’ relief alone.
• Roll. In hpr a pod adds no rolling moment of its own (but see A single pod’s moments below); it damps a roll. Rolling at p, a part at distance ρ from the axis crosses the air at p ρ, so it meets it at the angle p ρ/V, and its normal force acts about the axis with the arm ρ. That gives, for each pod’s body part, a roll damping (Roll: forcing and damping) of C_lp = −2 C_Nα ρ²/d², on the reference diameter d. A pod’s fins damp as the airframe’s fins do, with each strip’s distance taken from the rocket’s axis: a fin whose root is ρ_0 out along its span has its strips at ρ_0 + y.

Where the forces act. The flight engine applies each pod part’s force on the rocket’s axis at its station. For two pods or more, spaced evenly, the pods’ offsets add to zero, so the forces on the axis turn the rocket in pitch and yaw as they would at the pods. What an offset adds, to first order, is roll damping, which hpr adds as above, and a pitch damping from the pods’ drag: pitching at q, a pod ρ off the pitch plane meets the air q ρ faster or slower on either side, so its drag changes by as much and pitches back. That is C_mq ≈ −2 C_D,pod Σρ²/d² (the pitch moment per unit pitch rate, C_D,pod one pod’s drag coefficient, the sum over the pods), about −0.15 on the worked example below. hpr leaves it out: the airframe’s own pitch damping from its fins, −2 C_Nα,fins ℓ²/d² with ℓ the fins’ distance from the center of gravity, is of the order of −10³ on such a rocket.

Worked example. The tests’ rocket (the finned_rocket of hpr-aero’s tests: a 0.25 m ogive nose, 0.7 m of tube 27 mm in radius, a 0.05 m boattail to a 22 mm tail tube 0.3 m long, and four fins of 0.12 m root chord, 0.05 m tip chord, 0.06 m span and 0.07 m sweep) carries three pods 40 mm from its axis, starting 0.35 m aft of the nose tip. Each is a cone 0.05 m long on a tube 0.2 m long, both 10 mm in radius, so each pod’s fineness is 0.25/0.02 = 12.5. At Mach 0.3, Reynolds number 5×10⁶ per meter:

The pods move the center of pressure about 43 mm, 0.79 calibres, forward, and add 26% to the drag. The test the_aero_page_s_pod_example holds this table to the digits printed. The tests a_pod_adds_its_bodies_slopes_once_per_pod, a_pod_s_body_lift_takes_its_own_fineness, a_pod_drags_once_per_pod, a_pod_s_base_takes_its_own_motors_area, a_pod_s_bodies_damp_the_roll, a_pod_s_fins_turn_with_their_pod and a_pod_s_fins_are_the_pod_s_turned_with_it hold the rules to hand-worked values, the fins’ damping to Barrowman’s strips summed by hand over two pods with a fin pointing each way. In hpr-sim, pods_damp_the_spin_of_canted_fins_by_their_cones flies Valetudo with three pods and canted fins: it spins to the balance the pods’ damping predicts, to 1e-6.

Against OpenRocket. The six probe designs are written by pod_probes.py. Each is one airframe, a 0.2 m conical nose on 0.6 m of tube 60 mm across, with three fins and an AeroTech H128W, carrying one set of pods. The pods are about 0.3 m long and 20 mm to 32 mm across, 5 mm to 10 mm off the airframe; the winglets hang from pods of no length at its surface. Every outside part is set to OpenRocket’s regular paint finish, 60 µm roughness. OpenRocket 24.12 flies them straight up in calm air, and hpr flies the same files (hpr’s flights against OpenRocket’s). In such a flight the apogee and the largest speed test the drag and the mass; the normal force enters only through the margin, which both codes take at the moment the rocket leaves the rod. Pods’ change is the difference from pods-none, the same airframe with no pods, so it shows the pods’ own share in each code:

pods-motors-2 flies an H128W in each of its two pods and none in the airframe, twice the impulse, with 0.35 kg of nose ballast instead of 0.15 kg, so its change is not its pods’ alone. hpr’s largest speed is 0.59% to 1.24% above OpenRocket’s, its margin 0.0011 to 0.0014 calibres larger, and the pods’ change of margin agrees to 0.0004 calibres. The launch masses agree within 0.0001%. The test pod_designs_are_within_5_percent_of_openrocket holds every probe within 5% of OpenRocket’s apogee and largest speed, the bar every flight against OpenRocket is held to (ADR-076). It also holds each margin within 0.005 calibres, and the pods’ change within 1 percentage point of apogee and 0.005 calibres. Those bounds were set after these numbers were measured, tight enough to catch a broken pod rule: a pod fin’s body interference taken on the airframe’s radius would move a margin by some 0.04 calibres. The whole table is in the committed report.

The first probes stated no surface finish, and hpr flew them 6.0% to 7.3% high: OpenRocket reads a part with no finish as regular paint, 60 µm, and hpr’s reader gives it hpr’s own default, 20 µm (issue #216, which gives the run). The probes now state their finish.

One private design flies with pods too: C02, an anonymised design of the library (hpr’s flights of the private designs). Its pods hold only a launch lug each, on a tube of no length, so they add the lugs’ drag and nothing else. Over its 5 configurations hpr’s apogee is −0.94% to +2.15% from OpenRocket’s; 3 of them are compared with an OpenRocket flight whose parachute opened before apogee, and the +2.15% is one of those. Of OpenRocket’s own pod examples, Pods–airframes and winglets flies, its cockpit fin read on the nose cone (fins on a nose cone), with every apogee within 0.5% of OpenRocket’s and its margin 0.071 to 0.076 calibres above it, the flattering side (#325, #326). Pods–powered with recovery deployment flies since M4.5i, as saved since M4.5k, which weighs its rail buttons’ screw heads and takes each of its two pod sets’ motors off that set’s own pods (ADR-168). With only the sustainer’s motor it reads 0.36% below OpenRocket’s apogee, and with only the booster’s motors 1.86% below. Its staged flight reads 37.93% low: its sustainer is unstable when the air crosses it edge-on to its two-fin strake set, and in the conditions of OpenRocket’s record it turns over before apogee in both codes. That record’s site is at 28.61° N, where the Earth’s rotation tips hpr’s flight, and hpr’s angle of attack passes 90° at 2.25 s (hpr’s flights against OpenRocket’s).

What neither code models. OpenRocket documents no pod model of its own. The probes show that, on them, its answer is hpr’s: the pods’ change of margin agrees to 0.0004 calibres. Neither code turns the flow around the body, then: on pods-fins-2 the upwash beside the body (below) would add about a third to the pod fins’ force. So the limits below are both codes’ limits, and the agreement above cannot size them. The probes do not test a single pod (each has two or more), a pod faster than Mach 0.81, a rocket flying at an angle of attack (so not a pod’s body lift, which is zero straight into the air), airframe and pod motors burning together, or a roll: none has canted fins, and neither code’s roll is compared.

Left out, and how large it may be.

• The body’s flow around the pods, and theirs around it. The airframe turns the crossflow around itself, so a pod beside it meets the air at a different angle. In potential flow past a cylinder of radius a, a point at distance r, at the angle θ from the crossflow’s direction, meets it at α (1 − (a²/r²) cos 2θ) along the crossflow and −α (a²/r²) sin 2θ across it (at θ = 90°, NACA Report 1307’s upwash α (1 + a²/y²), [PNK57] p. 4 eq. 15). For three pods or more, spaced evenly, both average to zero, and the first-order change cancels. For one or two they do not: on the worked example’s pods, a²/r² = 0.46, so a pair’s cones lift up to 46% more or less, by roll angle, and push sideways by up to as much. hpr leaves this out; the probes above show OpenRocket leaves it out too. Interference drag is left out as well, as Barrowman leaves it out for fins ([B67] p. 62, “No interference drag effects are considered”) and Niskanen for the whole rocket ([N09] §3.4).
• A single pod’s moments. One pod’s drag acts off the axis and pitches the rocket; its normal force, and its fins’, act off the axis and roll it. hpr applies them on the axis, so it drops those moments (issue #213). On the worked example with one pod (drag coefficient 0.0445 at 40 mm) and a one-calibre margin, the pitch moment would trim the rocket at about 0.0445 · 0.04/(12.65 · 0.054) = 0.0026 rad, 0.15°.
• The interference on a pod’s fins’ roll damping. hpr takes the pod tube’s roll-damping factor k_R(B) on the whole sum. The part that comes from the pod’s offset is the pod moving sideways, whose factor is closer to the normal force’s K_T(B). For a fin whose span equals the tube’s radius (τ = 2), K_T(B) is 1.5 and a rectangular fin’s k_R(B) 1.33 (Fins, Roll: forcing and damping), so that part may be some 10% too small; no test sizes it. On a pod of no radius, winglets’ usual pod, both are 1 and nothing is lost.
• A pod inside the airframe. Nothing checks that a pod clears the airframe (#206); a pod sunk into it still gets its whole slope, friction and base.
• Canted fins on a pod are refused: their roll forcing about the rocket’s axis is not modeled. So is a pod’s tube of no length with a radius, a flat disc the drag buildup has no term for.

Tube fins

A tube fin set is a ring of short open tubes around the airframe, in place of flat fins. hpr flies each tube as an annular wing (a ring wing): a wing bent round into a tube, which lifts when the air meets it at an angle.

Only its parts are validated: no tube fin rocket has been checked against a wind tunnel or a measured flight. The ring wing’s slope is within 3% of five rings measured in a wind tunnel, which were thick and cambered, not paper tubes. What the tubes do to each other and to the body is not modeled: nothing measures it, and theory predicts more lift than hpr gives (#234).

The model came with M2.2e9 (ADR-099, tube fins flown as ring wings).

On OpenRocket’s Tube fin rocket hpr’s center of pressure is 1.07 calibres forward of OpenRocket’s. That misses the quarter calibre Loft lesson L19 asks for, and nothing measured says which code is nearer (below).

The normal force. A ring wing of diameter d and length L lifts about twice as much as a flat wing of span d and chord L. A long, thin ring lifts twice what a solid body of its diameter would, because it turns the air inside it as well as the air around it (Hoerner 1965, p. 7-13: L = q d² π α for a ring of small aspect ratio). hpr takes Weissinger’s formula for a thin ring (1955, as quoted by Wagner 2021, eq. 15), which runs from the short-ring limit to that long-ring limit. With λ = L/d, its lift slope reads as below. At small angles that is its normal-force slope, here on the area d L, not the reference area:

C_Lα = π² / (1 + πλ/2 + λ arctan(1.2 λ)) per radian.

d is the tube’s mean diameter, its outer and inner radii added; for a paper tube it is within a few per cent of the inner diameter the other choice would give. Fletcher’s rings run from λ = 1/3 to 3. A longer tube rests on the formula alone, which the tests hold to its long-ring limit. A set of N tubes takes N times one tube’s slope, on the rocket’s reference area. Faster than Mach 0, it takes Göthert’s rule, the same idea as the fins’ Prandtl–Glauert factor: the slope is the one a ring 1/β times longer would have, over β = √(1 − M²). That leaves the long-ring limit unchanged and scales the short-ring limit by 1/β.

The center of pressure. Fletcher measured where the lift of five rings acts (NACA TN 4117, 1957, Fig. 8, at Mach 0.13). hpr reads it against the ring’s aspect ratio A = d/L, interpolated in straight lines:

The table ends at A = 3, Fletcher’s shortest ring; a shorter ring is refused. Below A = 2/3 the line runs to the leading edge at A = 0. That end point comes from slender-body theory, in which a long, thin ring’s lift all appears at its front edge. Hoerner and Borst assume the same of the air turned inside an open tube: that it turns “at or near the rim of the inlet” (Hoerner and Borst 1985, p. 19-16). They write that they “do not have suitable experimental results at hand” on how an axial duct changes a slender body’s lift and moment.

Fletcher’s A = 1/3 ring is left out, and that is a judgement. Its center sits ahead of its leading edge. Fletcher puts that down to its low aspect ratio: such a ring behaves more like a slender body of revolution than the others (p. 4). That this would not carry over to a paper tube is hpr’s inference, untested. His rings had a Clark Y section 11.7% of the chord thick, all of it outside a straight bore. At a chord of three bores that wall is 0.35 of the bore thick, so about two thirds of the ring’s frontal disc is wall, against a few per cent for a paper tube. His thinner rings may carry some of the same forward shift. The choice matters. On the worked example below, the margin at rod clearance is:

The 0.29 holds Fletcher’s −0.11 below A = 1/3, as a first version of this model did; it comes from a one-off run, not kept in the report.

The drag. Friction takes the inside and the outside of every tube, 2π L (r_o + r_i) per tube, at the rocket’s skin-friction coefficient (Drag). The wall’s front ring, π (r_o² − r_i²) per tube, takes a square fin edge’s pressure drag: a blunt face at the front and base drag behind (Drag, eqs. 3.90 and 3.92).

Refused. hpr refuses these cases rather than guess:

• Mach 0.8 and faster. No source covers tube fins near the speed of sound, where the flow through a tube can choke. A flight that reaches it stops with the tube-fin model’s error, even on an override table, which still takes the tubes’ stations and roll from the model.
• Fewer than three tubes. The body’s own flow around three or more evenly spaced tubes cancels in the sum, but around one or two it doesn’t.
• Solid tubes, with a wall as thick as the radius.
• Tube fins on a pod.
• A ring shorter than a third of its diameter (A > 3), past Fletcher’s rings.
• Tubes that overlap each other.

A tumbling airframe with tube fins stays refused too (Recovery).

Worked example. OpenRocket’s Tube fin rocket has six tubes 76.2 mm long, of radius 12.395 mm with a 0.330 mm wall, on a body of the same radius. Its tubes are just longer than Fletcher’s longest ring, and their center falls on the line below his A = 2/3, the judgement above.

Against OpenRocket. The Tube fin rocket is in the flight report. Flown on hpr’s own drag, its apogee is 6.95% above OpenRocket’s, 302.5 m against 282.8 m. On OpenRocket’s recorded drag, hpr’s apogee is within 0.03% of OpenRocket’s (ADR-097). So the net gap is the drag’s, though parts of it could cancel, and the rest of the flight agrees. OpenRocket’s per-component drag, read through its public API but not kept as a record, points to where. These are leads for #228, not measurements the repository reproduces:

• The nose, the body and the lug agree within 0.005.
• The tube fins take 1.18 in OpenRocket’s total and 1.05 in hpr’s.
• Keeping the whole base while the motor burns, as OpenRocket does, closes 2.3 of the 6.95 points. This one is in the report: hpr flew it. That rule is not specific to tube fins (#222).

hpr’s tube-fin drag probably reads low. OpenRocket refined its tube-fin drag against measured flights of two tube-fin rockets. A table in a comment on the change that last revised it (openrocket#2066) has seven flights in five motor cases: OpenRocket’s apogee is within −2% to +5.2% of the measured one, and high in four of the five. hpr has checked itself against no measured flight, and a code-to-code gap is not a measurement. Still, that is evidence OpenRocket’s drag is the nearer of the two here. What hpr’s leaves out is listed below, and #228 holds the search for a measured source.

The center of pressure against OpenRocket. The stability margin differs more. At rod clearance, the moment the rocket leaves the launch rod, hpr’s margin is 0.79 calibres and OpenRocket’s 1.87. Almost all of that gap is the center of pressure: hpr’s is 1.07 calibres forward of OpenRocket’s, and the two centers of mass differ by 0.002 calibres. L19 asks for a quarter calibre, and hpr does not meet it (ADR-102, tube fins’ center of pressure measured against OpenRocket).

OpenRocket 24.12’s answers for tube fins are kept as a fixture, openrocket-tube-fin-aero.json, written by tube_fin_aero.py. It holds OpenRocket’s slope and center for the tubes and for the whole rocket, at five Mach numbers. It covers the Tube fin rocket and 14 probe designs, each changing one thing: the tubes’ length, count, radius or wall. The test tube_fin_cp_against_the_oracle_measured_and_pinned reads the same probes with hpr and pins all 70 gaps. The record shows:

• The two codes agree on the probes’ geometry and on the nose and body. Given OpenRocket’s slope and center for the tubes, hpr’s rocket has OpenRocket’s center of pressure within 0.01 calibres. So the whole gap is the tubes’.
• OpenRocket’s tubes lift 1.26 to 1.86 times what hpr’s ring wings do, at every Mach number.
• Its lift per tube is the same whatever the number of tubes, so it models no interference that changes with the count. Slender-body theory with the body included predicts one, and it falls as tubes are added. An unchecked estimate, taken 0.005 radii from the body and still rising as that gap closes, gives at least 1.96 times the lift of the same number of isolated rings for three tubes of 6 mm radius, 1.57 for six, and 1.13 for six touching each other (#234). OpenRocket’s lift on the 6 mm probes is 1.73 times the long-ring limit of isolated rings: below the estimate for three tubes. For six, the same estimate taken closer, and carried on to contact, is about 1.65, a little below OpenRocket’s. hpr leaves the interference out.
• OpenRocket puts the tubes’ center a quarter of their length aft of the leading edge up to Mach 0.5. Its maintainers describe that as the subsonic rule its flat fins and tube fins share (openrocket#3262).

On the probe built like the Tube fin rocket, six tubes 75 mm long touching the body and each other:

The Tube fin rocket itself, at rod clearance (Mach 0.056), gives −1.07: its tubes are 76.2 mm long and its nose and body differ a little from the probe’s.

From Mach 0.6 the gap shrinks only because OpenRocket 24.12 moves the tubes’ center to the leading edge. Its maintainers call that jump a bug and fixed it after 24.12 (openrocket#3235). The jump moves OpenRocket’s own center of pressure on the Tube fin rocket 0.73 calibres forward between Mach 0.5 and 0.6. It probably accounts for the five probe results, of 28 from Mach 0.6, that come within the lesson’s quarter calibre.

What each of OpenRocket’s two terms is worth: on that probe at Mach 0.05, giving hpr’s tubes OpenRocket’s center but keeping hpr’s slope moves hpr’s center of pressure 0.50 calibres aft. Giving them OpenRocket’s slope but keeping hpr’s center moves it 0.53 calibres aft. Up to Mach 0.5, every probe’s gap is between 0.42 and 3.0 calibres.

hpr keeps its model. Fletcher’s measured center moves forward, as a share of the ring’s length, as a ring gets longer, and Hoerner and Borst assume an open tube’s inner flow turns at its inlet. Nothing measured supports OpenRocket’s quarter length or its extra lift. hpr’s own center for this rocket is unmeasured too: its tubes (A = 0.32) are longer than every ring Fletcher measured. With his A = 1/3 ring left out, their center comes from hpr’s line from A = 2/3 to the leading edge. Placed as his thick A = 1/3 ring’s instead, it would give a margin of 0.29. Neither code has been checked against a measured tube-fin rocket, so which margin is nearer is open (#228). Until then, check a tube-fin design in both programs and treat the smaller of the two margins as the more cautious estimate. It is not a bound: the thick-ring reading above gives a smaller one still.

What it leaves out:

• How the body and the tubes change each other’s flow. Slender-body theory predicts that they raise each other’s lift, but hpr applies no interference factor (#234).
• The gaps between tubes and body, and the drag where they meet.
• How the flow through a tube develops, or chokes.
• A thin tube’s measured center of pressure.

The rocket’s own crossflow, R²/s² around a tube s from its axis, turns with twice the roll angle. It sums to zero over three or more tubes evenly spaced, which is why the model takes three or more. That is a first-order derivation, not a measurement: it takes the body’s flow at each tube’s center and leaves out the rest of their effect on each other.

The normal force from RASAero II

A flight can use another program’s normal force and center of pressure in place of hpr’s own. Today that program is RASAero II, read from the table it exports. Use it to fly two programs on the same aerodynamics, so that a difference between them comes from something else. Or use it to fly RASAero II’s numbers faster than sound, where hpr’s own normal force is less tested (Fins through Mach 1).

How far to trust it:

• The reading matches the file. On the export for Calisto, every one of its 4,999 rows at 2° and 4° comes back from hpr’s table to 2e-16. The 0° column, which hpr works out, agrees at 15 Mach numbers with the reading made for M1.8a.
• The flight uses the table as the equations say it should. A rocket flying on a table swings in pitch and yaw as the small-angle equations of motion predict for the table’s slope and center of pressure.
• Only up to Mach 0.75 on real data. Only one real export has been flown, Calisto’s. Faster than that, the table is checked by unit tests alone.
• Past the export’s last angle, and at 0° faster than Mach 1.3, hpr assumes. The assumptions fit RASAero II’s viscous part through Mach 1.3. Faster, that part grows much more slowly with the angle, so past 4° the table probably gives too much force at Mach 3 and above.

Three parts are hpr’s choices, not RASAero II’s:

• the damping, which stays hpr’s own;
• the normal force past the export’s largest angle of attack (4° in the one export tested);
• the slope at 0°, where the export’s normal force is zero.

In code, two calls take a file to a flight:

1. NormalForceTable::from_rasaero_csv reads the text.
2. Simulation::with_normal_force_table flies it. Its documentation has a worked program.

The decisions are in the record on normal-force overrides, ADR-032. The milestone is M1.8d.

What the export holds

RASAero II’s Aero Plots screen exports a table to CSV (File, Export, To CSV File; [RAS] p. 76). There is one row for each Mach number and angle of attack (Alpha, in degrees). The Calisto export has rows at 0°, 2° and 4°. hpr reads five of the columns:

CN also holds a viscous part, CN Viscous. It is the extra push, from the air’s friction, of the air flowing sideways across the body. RASAero II takes it from Jorgensen’s method ([RAS] p. 55), adds it from Mach 0.91 in Calisto’s export, and moves the center of pressure forward with the angle. hpr’s own model has neither. From Mach 0.91 through Mach 1.3 the viscous part grows exactly as sin² α: at 4° it is (sin 4°/sin 2°)² = 3.995 times its value at 2°. Faster, it grows more slowly: 3.90 times at Mach 1.5, 3.16 at Mach 2, 1.73 at Mach 3 and 1.05 at Mach 4. The export’s CNalpha (0 to 4 deg) and CP (0 to 4 deg) columns repeat its 4° values on every row; hpr doesn’t read them.

How hpr reads it

hpr builds one column for each angle in the export. Each column holds C_N/α (the normal force over the angle, per radian) and the center of pressure, both against Mach number.

• At a positive angle, C_N/α is CN over the angle in radians.
• At 0°, CN is zero, so it can’t be divided. hpr takes CN Potential at the smallest positive angle, over that angle. The potential part grows in step with the angle: in Calisto’s export its C_N/α is the same at 2° and 4° to 2e-15. Through Mach 1.3 the viscous part grows as sin² α, so it adds no slope at 0°. Faster, the export doesn’t show how it starts from 0°, and leaving it out of the slope is an assumption.
• The center of pressure is converted from inches to meters at 0.0254 m to the inch. It stays measured from the nose tip, as hpr’s stations are (station). So the design must start at the same nose tip as the RASAero II file. A table whose center of pressure, at one of its Mach numbers up to 5 (where a flight stops), lies outside the rocket, ahead of its nose or behind its tail, is refused: it is the sign of a length in the wrong unit.
• Reference area. RASAero II’s coefficients are on the body’s largest cross-section ([RAS] p. 72). hpr records that and rescales them to the rocket’s own reference area, when that is something else. For a rocket of several stages, export the whole stack (“Sustainer plus Booster” or “All Stages”): hpr flies the whole stack.

A flight looks up the table at its Mach number and angle of attack:

• Within one column, the values are linear in Mach number. Outside a column’s range the end values hold, and the lookup says so.
• Between two columns, C_N/α and the center of pressure are linear in the angle. Then C_N = (C_N/α)·α comes back exactly at each column’s angle. Between them it is a part in step with the angle plus one in its square: RASAero II’s own shape through Mach 1.3 (α² is within 0.2% of sin² α to 4°), and an assumption faster than that.
• Past the largest angle α_n, the normal force splits in two. The linear share is the slope at 0° times α_n, at the 0° center of pressure; it grows as sin α, as hpr’s own fins do (Aerodynamics in flight). The rest of the force, with the rest of the moment, grows as sin² α, the form of the air crossing the body that hpr’s body lift also takes ([G] p. 1; [N09] eq. 3.26). The force and center of pressure are continuous at α_n, and the force is zero when the air comes from the tail. Two limits keep this sensible for any table. The rest’s center of pressure is held within the rocket. And a table whose C_N/α falls with the angle has no rest: its whole force grows as sin α, so the force never turns round. Neither limit makes a jump as the Mach number changes. Either way this is an assumption, and the lookup reports it.

A worked example. Take an export with invented numbers. At Mach 1, CN Potential is 10 per radian times the angle, CN Viscous is 0.03 at 2° and 0.12 at 4°, and the center of pressure is 50, 49 and 48 inches at 0°, 2° and 4°. These are the numbers in the CSV reader’s unit test, reads_a_rasaero_export_by_angle_of_attack.

At 10°, sin 10°/sin 4° is 2.4893. The linear share, 10 × 0.069813 = 0.69813 at 1.2700 m, grows to 1.7379. The rest, 0.12 at 0.9237 m (the station that gives the 4° moment), grows by 2.4893² to 0.7436. Together they make 2.4815 at 1.1662 m: the center of pressure moves forward with the angle, as RASAero II’s does. The first draft scaled the whole 0.81813 by 2.4893 at 1.2192 m instead: 2.0366, 18% less force, with the center of pressure 5.3 cm further aft. That reads the rocket as more stable than the viscous part makes it.

In a flight

The flight takes the table’s normal force at the center of mass’s airflow and applies it at the table’s center of pressure.

The export has no damping, so hpr keeps its own (Rigid-body flight). The table gives the force as if the rocket weren’t turning. When it turns, each part of the rocket meets the air at a slightly different angle, and hpr adds that difference: it is the damping. When the rocket isn’t turning, the difference is exactly zero.

The flight still stops at Mach 5, where hpr’s own parts, which give the damping, end. The table gives no side force: RASAero II’s rockets are symmetric.

How it was checked

The Calisto flights show how much the change matters. They are not a check of accuracy: nothing measured flew. cargo xtask aero writes the fixture from the export, which isn’t committed, and checks it again wherever the export is present. CI has no copy of the export: there, a test checks that the fixture’s 0° values agree with the ones committed for M1.8a, and flies the 15-point table again.

What it leaves out:

• No real export has been flown through Mach 1. Calisto peaks at Mach 0.75. The reader’s unit tests cover the transonic columns.
• Past the export’s largest angle, the split continuation is hpr’s assumption. In a 5 m/s crosswind Calisto flies past 4° for about 0.3 s just after leaving the rail (up to 7.9°) and for the last 1.9 s before apogee, as it slows below 30 m/s. Its export has no viscous part below Mach 0.91, so Calisto’s flights use only the linear share; the tables of hpr’s own normal force are the flights that grow a rest as sin² α. From Mach 3, where RASAero II’s viscous part hardly grows between 2° and 4°, the sin² α share probably gives too much force.
• The nose tip can’t be checked from the export beyond the refusal above. A design that starts somewhere else gets a shifted center of pressure, with no warning.
• Only RASAero II’s layout is read. A table from anywhere else can be built in code with NormalForceTable::new.

Drag

Code: hpr_aero::drag (the terms), AeroModel::drag and AeroModel::buildup_components (their sum over a rocket), hpr_aero::table (override tables), hpr_design::Finish (roughness). Decisions: ADR-009 (drag buildup, surface finishes and override tables). Extra sources:

• [B67] ch. 4 (pp. 43–62): the friction, roughness and leading-edge formulas Niskanen adopts, and Table 4-1 of roughness heights (p. 46, after Hoerner p. 5-3).
• [N09] §3.4 (pp. 41–53) and appendix B (pp. 106–110). [TD] reprints the same drag equations and tables unchanged.

On this page C_D0 is the zero-lift drag coefficient: the drag with the air straight along the axis (no angle of attack), divided by q A_ref. Recovery uses the same symbol for something else: a parachute canopy’s drag coefficient on its nominal area.

Drag is built up term by term. Skin friction acts over the whole surface. Pressure drag acts on noses and shoulders (here, a transition that widens toward the tail) and on boattails (one that narrows). Base drag acts on the flat aft end, fin pressure drag on the fins’ edges, and parasitic drag on launch lugs and rail buttons: C_D0 = C_D,friction + Σ_T (A_T/A_ref)(C_D•)_T ([N09] eq. 3.75, 3.97), each pressure, base and parasitic term on its own area. The axial coefficient is C_A = C_D0 f(α).

• Roughness. hpr_design::Finish names the fifteen rows of [B67] Table 4-1 (0 to 1000 µm; [N09] Table 3.2 reprints ten) or takes a custom height. The default is “paint in aircraft mass production”, 20 µm. Each component has its own finish; the Reynolds number and R_s/L use the whole rocket’s length, as [N09] does (Loft lesson L12). Loft cited none of its values: its 60 µm is OpenRocket’s “regular paint” ([N09] p. 83), its 2 µm isn’t in either table, and its 1 + 60/f³ + 0.0025f is Raymer’s aircraft fuselage form factor, not [N09]’s.
• Fully turbulent. The boundary layer (the thin layer of air the skin drags along) is taken as turbulent everywhere, never laminar (smooth and layered). [N09] p. 43 found laminar runs changed apogee by under 5% and dropped them. Eq. 3.81’s R < 1e4 branch applies first, even on surfaces rough enough that R_crit < 1e4.
• Friction jumps where [N09] does. Eq. 3.79 is not where eq. 3.78 and 3.80 cross, so eq. 3.81 jumps at R_crit: +9% for 60 µm on a 1 m rocket (0.00419 to 0.00458). The subsonic and supersonic corrections also differ at Mach 1 (0.900 against 0.922 turbulent). hpr keeps the published forms, and the tests pin both jumps (Loft lesson L90).
• Friction on the axial projection (a departure). Wall shear (the air’s drag on the skin, τ per unit area) acts along the surface, so on an element of area dA at an angle θ to the axis its axial share is τ cos θ dA, and the body’s friction area is 2π ∫ r dx = π A_plan rather than the slant surface in [N09] eq. 3.85. On slender noses the difference is small: a tangent ogive loses 1.1% of its own friction area at fineness 3 and 2.4% at fineness 2. On a short, steep shoulder it removes friction on what is nearly a flat face, so a shoulder’s drag tends to a bare step’s as its length goes to zero (Loft lesson L15); with the slant surface it would stay about C_fc ΔA/A_ref above it. The OpenRocket comparison (M2.2) will measure the difference.
• Steps in radius. Where one body component meets the next with a different radius, a step up is a zero-length shoulder, a flat face (0.8 ΔA at rest, rising with Mach as under Drag through Mach 1), and a step down a zero-length boattail, the base drag of the uncovered area. A body with no nose cone gets the same flat face on its front. Each is the limit of the transition it replaces (Loft lesson L15), and it is reported with the aft component. A step up just behind a step down, such as a motor retainer behind the step to the motor tube, is sheltered by the step’s corner at every speed, by how far it rises and how much motor tube shows ahead of it, and not at all once that is as long as the step’s drop in diameter. This is unmeasured; for a 98 mm airframe with 12 mm of a 54 mm motor tube showing and a 62 mm retainer it lowers the rocket’s C_D0 12% to 23% from Mach 0.3 to 2.5 (Boattails faster than sound).
• Boattails. [N09] eq. 3.88 writes A_base/A_boattail without defining the areas, and p. 48 says a zero-length boattail drags like “the total base drag”. Taking A_base as the aft base would count that base twice and leave out the uncovered ring (annulus), so hpr reads both as the boattail’s decrease in area (Calisto’s boattail: 0.052, against 0.046 the other way). The joint angle is atan(dr/dx) at the aft end, ±π/2 where a curved transition ends in a blunt tip. A lip (a short step up or flare) just behind a boattail or a step down is in its wake at every speed, and from Mach 0.8 a boattail’s drag rises to its supersonic wave drag (Boattails faster than sound).
• Base drag under power subtracts the thrusting motors’ cross-section from the aft base, down to zero ([N09] pp. 50–51: eq. 3.94 on p. 50, and on p. 51, “if the base is the same size as the motor itself, no base drag”, which Niskanen takes from Fleeman’s Tactical Missile Design; Loft lesson L13). Neither this rule nor OpenRocket’s below has been checked against a measured flight. On one private design’s supersonic flight, C06/1, the choice moves hpr’s apogee difference from OpenRocket’s from +13.60% to −10.71%, about 24 percentage points, more than any other known cause (#222, open on both this and the supersonic pressure drag). DragConditions::thrusting(reynolds_per_m, motor_area_m2) takes the cross-section of the burning motors from the flight engine (zero when unknown: no relief). The base belongs to the last body component.
  • OpenRocket’s rule, for comparisons. OpenRocket 24.12 does not take the motor off. While a motor burns, its base drag is the whole base’s. A committed probe measures this: validation/oracles/openrocket/base_drag.py records OpenRocket’s base drag on its own example designs, the rows while a motor burns against the rows after, in validation/fixtures/ork/openrocket-base-drag.json, and a test in hpr-validate holds it (ADR-097, the decision on sizing a drag cause). On all 42 of its flights of one data branch (nothing separating), the base drag while a motor burns is exactly the whole base’s, to 1e-12, where the motors cover 9% to 94% of the reference area, one flight’s motors in pods. On a rocket whose motor fills most of the base, the rule is a large difference under power. On C06/1, switching hpr to OpenRocket’s base rule alone lowers its apogee from +13.60% to −10.71%, 24.3 percentage points. Switching the rest of the drag to OpenRocket’s then raises it to +1.11%, 11.8 percentage points. The second number is by subtraction, and the split depends on which change is made first (a supersonic flight). AeroModel::with_full_base_drag_under_power() and Simulation::with_full_base_drag_under_power() fly OpenRocket’s rule. A sustainer lit after a powered separation keeps it. They exist to size a difference from OpenRocket, not as a better model: hpr keeps Niskanen’s rule of taking the motor’s area off the base by default.
  • Supersonic pressure drag against OpenRocket. Faster than sound, the pressure on the nose, the fins’ edges and any step is mostly wave drag. On that flight hpr’s supersonic pressure drag is about twice OpenRocket’s: OpenRocket gives the nose almost none well above Mach 1, and the fins about a quarter of hpr’s. Which is right is open, since neither has been checked against a measurement on that shape (#222: hpr’s supersonic pressure drag is about twice OpenRocket’s).
• Fins. Each fin set is its own term with its own thickness, chord and cross-section, so their order doesn’t matter (Loft lesson L11). c̄ is the mean aerodynamic chord and Γ_L the leading-edge sweep: atan(x_t/s) for a trapezoid, the span average for freeform outlines ([N09] p. 50), and for an ellipse π/2 − acos(k)/√(1 − k²), k = c_r/(2s) (the closed-form average, with its acosh form for k > 1). The drag goes as cos² Γ, whose span average is 6% lower than cos² of the average angle for an ellipse of k = 1. Fin–body interference drag and tip vortices are neglected ([N09] p. 41).
• Launch lugs. d in eq. 3.95–3.96 is taken as the outer diameter: [N09] p. 52 treats a solid rail pin as a lug “with a length equal to its diameter”, which only reads that way (Loft lesson L14). A row of count lugs is count lugs. Rail buttons follow [N09]’s rail-pin rule on their side profile (base and flange at the outer diameter, waist at the inner).
• Angle of attack (derived coefficients). [N09] §3.4.7 describes, without an equation, a two-part polynomial from 1 at 0° to 1.3 at 17° and 0 at 90°, with zero slope at each. hpr uses the unique cubic on each part that meets those four conditions. C_A is positive toward the tail. Past 90° the flow meets the tail, and hpr mirrors with the sign reversed, −f(180° − α), an assumption that keeps drag opposing the motion. The planned OpenRocket comparison (M2.2) will check it against OpenRocket, whose polynomial may differ.
• Refusals, not clamps (Loft lesson L16). Geometry the terms can’t use (a lug wall thicker than its radius, a button’s base and flange taller than the button, a negative roughness, which Rocket::layout already refuses) is an error naming the component; a coasting condition (no motor burning) with a motor area and a non-finite result are errors; large coefficients are returned as they are.
• Override tables (DragTable) replace hpr’s own C_D0 with curves of C_D0 against Mach number from another tool, power-off and power-on, read from CSV text: two columns, optionally under a header (RocketPy’s curves; \r\n, a byte-order mark and 01.05 accepted), or a header naming the column, with rows at non-zero Alpha skipped (RASAero II’s export: Mach, Alpha, CD, CD Power-Off, CD Power-On, …). An identical repeated row is skipped; a Mach number repeated with another value, or out of order, is refused with its line, not sorted. Tables interpolate linearly and hold their end values; Drag::table reports any extrapolation. DragConditions::thrusting selects the power-on curve. A table’s reference_diameter_m, when set, rescales it to the rocket’s reference area. The angle-of-attack factor still applies, and an override accepts any Mach number. AeroModel::buildup_components always reports the buildup, table or not. The normal force has a table of its own (The normal force from RASAero II).

A part’s stated drag coefficient

A part’s or stage’s stated drag coefficient flies as OpenRocket 24.12 flies it. The change it makes to the rocket’s drag is within 0.0059 of OpenRocket’s on 25 probe designs, but for lugs and buttons (up to 0.072, hpr’s own lug and button drag being higher) and, faster than Mach 0.6, for a nose or a boattail. This checks the rule, not the rocket’s whole drag, and the stated number itself is the designer’s.

OpenRocket lets a part say its own drag coefficient, in place of the drag its shape gives (the Override tab’s coefficient of drag, saved as <overridecd>). Designers use it for a part whose drag they measured, and for “hacks” such as OpenRocket’s Base drag hack example, which hangs a weightless, dragless cone behind the rocket to make OpenRocket count a second base. hpr reads the setting from a .ork into DragOverride, a part’s or stage’s drag_override. OpenRocket’s documentation names the checkbox but not what it replaces, so the rule below was measured on 25 probe designs (drag_override.py; ADR-167, a part’s drag override as OpenRocket flies it).

• The coefficient is on the rocket’s reference area, the same at every Mach number, and counted once per instance: per fin of a fin set, per lug or button.
• It replaces all of the part’s own drag: friction, pressure drag, and the base drag of its aft face, both the rocket’s base and a step down to a narrower part behind it. A step up at its fore end is its own face too. A step down just ahead of it stays: that is the aft face of the part in front.
• With override for all subcomponents (the design’s include_children), the parts attached to it have no drag of their own.
• On a part inside the body (an inner tube, a ring, a mass) it does nothing; such a part has no drag anyway. On a stage it is added to the rocket’s drag, or, covering its children, it is all of the stage’s drag.
• hpr refuses by name what OpenRocket wasn’t measured on: a coefficient on a pod set, on a part in a pod, on a tube fin set, or covering a pod set.

A worked example. OpenRocket’s Base drag hack (short-wide) is a nose, a body tube 78.7 mm across, and a 247 mm cone behind it that widens from a point back to the tube’s diameter. The cone is stated at 0. hpr’s C_D0 at Mach 0.3, from the example file:

Checked against OpenRocket. On every probe, the change in the rocket’s drag from the same airframe with nothing stated is held to OpenRocket’s (a_stated_drag_coefficient_moves_the_drag_as_openrocket_s). The stated number is the same in both, so what differs is each code’s own drag of the parts it replaces.

Faster than Mach 0.6, the two codes’ own drag of a nose and a boattail differs by up to 0.21 and 0.061; that is the drag buildups’ difference, not the setting’s.

What it leaves out. When an aft part is stated, OpenRocket also raises the friction of the parts left, by 6.8% of their friction on the Base drag hack example (about 1% of its C_D0), as though the body were shorter. hpr keeps each part’s friction as its shape gives it: the parts are all still there.

On the example’s flights. With recovery in both, Base drag hack reads +1.95% and +4.52% on OpenRocket’s apogee on a C11-5 and a D12-3 motor, and +7.80% on an E12-4. Flown on OpenRocket’s own drag curve, hpr comes within 0.1% of OpenRocket’s flight with nothing deployed, so what is left is the drag coefficient: by OpenRocket’s breakdown, the stubby ellipsoid nose (0.58 calibres long). OpenRocket gives it 0.064 at Mach 0.3. hpr gives it 0.0008, read off a straight line between Hoerner’s two measured round heads, a hemisphere at 0.01 and a longer head at −0.05 (blunt ellipsoids). A one-off probe, not committed, found that about 0.013 to 0.015 would bring the E12-4 within 5%, more than the hemisphere’s 0.01. The gap stays, and the 5% bar is not met. Nothing here measures which program’s nose drag is nearer the truth, but the measurement puts a head this long at or below the hemisphere’s (ADR-173, a blunt ellipsoid’s measured drag; #177).

Drag through Mach 1

Near the speed of sound a nose starts to push shock waves ahead of it, and the pressure on its surface climbs: the transonic drag rise. Past Mach 1 this pressure drag, called wave drag, settles to a value set mostly by the nose’s shape and how slender it is. hpr follows Niskanen’s method ([N09] §3.4.3 and appendix B, pp. 47–48 and 106–110) for noses, shoulders and steps. The other terms already had their faster-than-sound forms in the table above: friction’s Mach correction, base drag’s 0.25/M, the fins’ leading and trailing edges, and the stagnation pressure on blunt faces. The code is hpr_aero::nose_drag; the decision record is ADR-028. How far it holds against a wind tunnel is under Verification: it reads high except near Mach 1, most of all with fins on past Mach 1.2.

A nose’s or shoulder’s pressure-drag coefficient, on the area it adds, has three parts:

• At rest, eq. 3.86’s 0.8 sin² φ, with φ the joint angle at the aft end (the table above). A step, or a bare front face, has no length, and takes the flat face’s 0.85 q_stag/q at every Mach number instead (below).
• From M_L, the Mach number where the transonic formula takes over (0.8 or 1 by shape, in the table below; an ellipsoid has its own curve at every Mach number), a transonic and supersonic value C_T(M) that depends on the shape and the fineness ratio f = l/(d_aft − d_fore): a nose’s length over its base diameter, and for a shoulder its length over its rise in diameter, so that a conical shoulder drags like the cone with the same surface angle.
• Between rest and M_L, eq. 3.87: a Mᵇ + 0.8 sin² φ, with a and b chosen so the curve meets C_T and its slope at M_L: b = C_T′(M_L) M_L/Δ and a = Δ/M_Lᵇ, where Δ = C_T(M_L) − 0.8 sin² φ. Niskanen asks for a curve that doesn’t fall and is flat at rest, which needs Δ > 0 and b > 1; otherwise hpr uses a quadratic (below). With b near 10, as for slender cones, the curve stays close to its value at rest until about Mach 0.8.

Here γ = 1.4 is the ratio of specific heats of air, q_stag/q the stagnation-pressure ratio of the table above, and the fineness scaling is the curve a/(f + 1)ᵇ through a flat face at fineness 0 and the measured nose at fineness 3 (eq. B.7). Stoney’s report suggested that fineness and Mach number act separately ([N09] p. 108).

Stoney’s curves. Stoney’s 1961 NASA report collected the drag of about 200 bodies flown on rockets at NASA Langley ([S61]). Its Figure 12 plots the pressure drag of noses of fineness 3 against Mach number: panel (a) from flight models, Mach 0.8 to 2.0, and panel (b) from a wind tunnel (his ref. 30), to Mach 3.6. No table prints them, so hpr carries them as points read off a 600-dpi scan of the figure, each panel’s grid calibrated where the curve runs, to about ±0.0015. hpr takes panel (a) for the seven shapes it has, and panel (b) for the x^¼ and the ellipsoid, which only it has. Where (a) and (b) overlap, (b)’s von Kármán reads 0.004 to 0.011 higher from Mach 1.2. Past a curve’s last point hpr holds its last value. Panel (b) checks two of those holds: its von Kármán reads 0.079 to 0.086 from Mach 2.4 to 3.59, against panel (a)’s held 0.079, and its x^¾ falls to 0.073 by Mach 3.2, 8% under the held 0.079; panel (a)’s x^½ is still rising at its end. The x^¼ and the ellipsoid, which panel (b) starts at Mach 1.2, are joined by a straight line to 0 at Mach 0.8, where every smooth 3:1 nose of panel (a) reads 0; so every measured shape starts at Mach 0.8. An elliptical nose of another fineness draws its own line, after the scaling, from its low-speed value, held to Mach 0.8 (blunt ellipsoids, below). The points and where each was read are in the code (StoneyNose). A sample, on the nose’s base area (panel (a)’s values at Mach 3.0 are its held end values):

A shape between two measured ones interpolates between their curves in its parameter (the exponent n, K′ or C of Shapes), before the fineness scaling ([N09] p. 108): a power series xⁿ runs through a flat face (n = 0), x^¼, x^½, x^¾ and the 3:1 cone (n = 1); a parabolic series through the 3:1 cone (K′ = 0) and the ½, ¾ and full parabolas; a Haack series between von Kármán (C = 0) and L-V Haack (C = ⅓).

Worked example. A 5:1 von Kármán nose at Mach 1.5. Stoney’s 3:1 von Kármán gives 0.0893. The flat face gives 0.85 q_stag/q = 0.85 × 1.5381 = 1.3074, with q_stag/q = 1.84 − 0.76/1.5² + 0.166/1.5⁴ + 0.035/1.5⁶. The exponent is log₄ 6 = 1.2925, so the nose drags 1.3074 × (0.0893/1.3074)^1.2925 = 0.0407 on its base area. A 5:1 cone drags 0.0653 there, from eq. B.4, so the von Kármán’s wave drag is 38% lower. The test nose_drag::tests::the_guides_worked_example pins these numbers.

Where hpr departs from, or adds to, the source (ADR-028):

• Short cones and ogives. Below fineness 1 the cone formula runs past a flat face’s drag: as the cone flattens, eq. B.4 tends to 2.39 at Mach 2 against the flat face’s 1.41. So below fineness 1 hpr scales, at every Mach number, between a flat face at fineness 0 and the whole curve of the cone at fineness 1, as eq. B.9 scales the measured shapes. A shoulder then tends to a bare step as it shortens (Loft lesson L15), and above fineness 1 Niskanen’s cone is unchanged.
• Steps. A step up in radius, or a body with no nose cone, is a flat face: the blunt cylinder’s 0.85 q_stag/q at every Mach number, 0.85 at rest, 0.9947 at Mach 0.8, 1.0888 at Mach 1 and 1.4118 at Mach 2. Eq. 3.86 “does not take into account the effect of extremely blunt nose cones (length less than half of the diameter)” ([N09] p. 47), and a step has no length. Before M1.8b1 it was eq. 3.86’s 0.8 at every speed.
• Blunt ellipsoids. An elliptical nose has its own curve below Mach 1.2, from Hoerner’s measurement (blunt ellipsoids, below).
• Where eq. 3.87 has no solution. An x^½ nose meets its tube at a small angle, so it has some drag at rest, but Stoney’s measured x^½ curve is still at 0 at Mach 0.8: no a Mᵇ can rise from the first to the second. Eq. 3.87 needs the transonic value above the value at rest and b > 1; where either fails, hpr goes from the value at rest to C_T(M_L) along 0.8 sin² φ + Δ (M/M_L)² instead: continuous, flat at rest, with a kink at M_L. Falling, as here, it follows Stoney’s measurement rather than Niskanen’s assumption that the curve doesn’t fall; the coefficients are below 0.01. Rising, it serves near-flat noses: a power series x^0.05 has almost nothing at rest by eq. 3.86, which leaves bluntness out, and rises along it to 0.80 at Mach 0.8.
• Refused shapes. A bulged secant ogive (its arc radius below the tangent ogive’s) is outside eq. B.8, and a Haack series past C = ⅓ outside Stoney’s data (Niskanen limits it the same way, p. 103). The drag buildup refuses both, naming the component, when asked for drag; the model still builds, so the normal force, the center of pressure and a drag table still work.

Blunt ellipsoids below Mach 0.8

This covers an elliptical nose’s pressure drag below Mach 1.2 (ADR-173, a blunt ellipsoid’s measured drag). Stoney’s ellipsoid curve is 0 at Mach 0.8, and eq. B.9 keeps it 0 at any fineness. So before, an ellipsoid nearly as blunt as a flat face got no drag below Mach 0.8, though a flat face gets 0.85. Hoerner measured the forebody pressure drag of round heads on a cylinder at low speed, from their pressure distributions ([H65] p. 3-12, Fig. 20), and hpr now uses it:

• A hemisphere (0.5 calibres long) reads 0.01, and a round head about one calibre long −0.05: the suction on its shoulder outweighs the stagnation pressure at its tip.
• Between them, a straight line, falling 0.12 per calibre past the hemisphere. It crosses 0 at 7/12 calibre, and hpr holds it there, since it never charges a negative pressure drag.
• Blunter than a hemisphere, eq. B.9’s form between the flat face (0.85 q_stag/q, 0.85 at Mach 0) and the hemisphere’s 0.01. This is hpr’s interpolation: nothing in Fig. 20 measures an ellipsoid between the two.
• From the hemisphere up, the value holds to Mach 0.8, and blunter heads follow the flat face’s rise with Mach number. That is hpr’s assumption: the measurement is at low speed, and the drag rise begins near the critical Mach number, about 0.65 to 0.7 for a hemisphere.
• From Mach 0.8, a straight line to Stoney’s curve, scaled to the nose’s fineness, at Mach 1.2, its first point. This is hpr’s join, not data. The scaling comes first, then the line; scaling the line instead, as before, made the drag rise with an infinite slope at Mach 0.8.

Worked example. A ¼-calibre ellipsoid at Mach 0: 0.85 × (0.01/0.85)^(ln 1.25/ln 1.5), where 1.25 is 1 + ¼ and 1.5 is 1 + ½ (the hemisphere), is 0.85 × 0.0867 = 0.074. Base drag hack’s 0.577-calibre nose lies on the line: 0.01 − 0.12 × (0.577 − 0.5) = 0.0008. The test nose_drag::tests::a_blunt_ellipsoid_takes_hoerners_measured_forebody_drag pins both.

How far to trust it. The values come from one figure, which gives no Reynolds number and no length for the round head; drawn 1.4 calibres long instead of one, it would give this nose 0.005. No measurement covers such a head between Mach 0.3 and 1.2.

Cross-check against a measured cone. Stoney’s Figure 12(a) also has a 3:1 cone. Niskanen’s closed form reads high through the whole rise: +87% at Mach 0.8 and +105% at 0.85, where eq. 3.87 carries its Mach 1 value down; +49% at Mach 1 and +48% at 1.1, where the cubic join is near its peak, 0.234 against the measured 0.158; then +15% at Mach 1.5 and +4% at Mach 1.94, the curve’s end (nose_drag::tests::niskanens_cone_against_stoneys_measured_cone). Ogives inherit this. So a stubby cone or ogive gains the most drag at high subsonic speeds: Bella Lui’s 1.55:1 tangent ogive takes its whole rocket’s C_D0 at Mach 0.9 38% above the model before M1.8b1, which held the nose at its value at rest, where the von Kármán noses of Calisto and Prometheus move it under 1% (ADR-028).

Boattails faster than sound

A boattail narrows the body toward the tail, usually to shrink the flat base behind it. Below Mach 0.8 hpr keeps Niskanen’s boattail rule from the table above, a share of the base drag. Faster than sound two more things happen. The air turns inward around the boattail’s shoulder and expands, as in a Prandtl–Meyer expansion: it speeds up, its pressure falls below the free stream’s, and it pulls back on the boattail. That is a wave drag, and on a short, steep boattail it can be the largest drag term on the rocket. Behind the boattail, the base’s pressure is higher than behind a plain cylinder, which lowers the base drag. Code: hpr_aero::afterbody. Decision: ADR-030.

How far to trust it. Against 58 readings of 20 measured boattails of 3° to 10° from Mach 1.2 to 3.12 it reads −21.9% to +28.3%, and within 0.0123 in drag coefficient: the largest percentages are the smallest drags. Theory that leaves out the air’s viscosity (inviscid theory) reads such boattails up to about 20% high ([CS51] p. 17). Through Mach 1 it reads low, and below Mach 0.8 the rule gives long, gentle boattails almost nothing. Steeper boattails in a thick boundary layer read 26% to 54% high, and the one full rocket measured with one, the Arcas Robin, reads high too. So M1.8b3’s targets were not met: none of the Arcas Robin’s 11 fins-off readings from Mach 1.5 is within 10%, and 8 of Calisto’s 17 supersonic rows against RASAero II are. No whole flight in the validation suite uses this model yet: none of its boattailed rockets passes Mach 0.8. The rules for a boattail drawn in parts, a lip in its wake and the gaps between (the last three rows of the table) apply at every speed, and are judgements that no measurement checks but the Arcas Robin’s lip.

In the table, a boattail runs from diameter d₁ to d₂ over its length l; a = (d₂/d₁)² is its area ratio, θ = atan((d₁ − d₂)/(2l)) its half-angle (the cone through the same ends; curved boattails are taken as that cone), and coefficients are on its fore area π d₁²/4, the cross-section where it starts. C_p,PM is the pressure coefficient after the Prandtl–Meyer turn, (C_D•)_base the base drag coefficient of the table above, a_b the base’s area over π d₁²/4, and p_cyl/p_bt a cylinder’s base pressure over a boattail’s. “Jet off” means the measurements were made with no motor exhaust.

Why each piece is there:

• The chart is second-order theory. MIL-HDBK-762 cites no source for it. Jack computed conical boattails by Van Dyke’s second-order theory ([J53]), an inviscid method that keeps the next term past linear theory’s small-disturbance approximation, from Mach 1.5 to 4.5, and the chart agrees with his 83 points inside it within −10.4% to +8.0% for a up to 0.6. First-order (linear) theory reads far higher at these angles, so the chart is not linear theory.
• The 2D limit matters for short, steep boattails. The chart plots its drag against one combined variable, x, which holds only for small angles, and near Mach 1 it can ask for more suction than a flat (two-dimensional) turn gives, which a round boattail, whose pressure recovers aft of the shoulder, can’t exceed. Past the chart the same limit gives the curve its shape: against Jack’s 28 points beyond it, −2.7% to +8.0%.
• Near Mach 1 no method exists for boattails; [762] p. 5-47 says so and advises holding the supersonic value to a peak between Mach 1.0 and 1.2. The straight line starts at Mach 0.8, where hpr’s other transonic terms start, and is half-way up at 0.9. The measured rise is later and steeper: half-way by about 0.89 for Compton’s 10° boattail ([C72]) and 0.92 to 0.96 for Cubbage’s ([C57]), with a peak at Mach 1.0 to 1.1 that holding the Mach 1.2 value doesn’t reach. An earlier draft started the line at Mach 0.9, a choice made after seeing the Arcas Robin, one of the targets; the validation audit caught it, and it was put back to 0.8.
• The base’s relief is the handbook’s correlation, measured at Mach 2.5 to 3.5. Used as a ratio of pressures below Mach 2.5 it over-predicts the relief of the bases measured at Mach 1.59 and 1.91 ([DN54], [CS51]); held as a ratio of coefficients, it matches them on average. It depends on the base’s area only, where the measured relief also grows with the boattail’s angle: behind Cortright and Schroeder’s small bases (a_b = 0.256) hpr keeps 0.352 of the cylinder’s base pressure coefficient at every angle, where they measured 0.60 at 5.6° and 0.26 at 9.3°.
• The lip. NASA’s Arcas Robin models end in a lip 1.3 mm long that flares from the boattail’s end to the base. NASA found it lowering the force on the balance chamber inside the base at Mach 1.5 and 1.8 with the fins off, and its effect “masked” when the flow over the boattail separates or the boundary layer thickens ([D4014] p. 6); RASAero II’s own comparison with the tunnel left it out as “buried in the boattail boundary layer” ([R22]). hpr used to take it as a stubby cone in undisturbed air, 0.085 of drag. The quarter and half that bound the wake were chosen knowing this lip rises 0.17 of its boattail’s drop, and every one of the 44 Arcas Robin rows depends on that choice: with the lip counted as a shoulder in undisturbed air, each would read 0.065 to 0.086 higher.

A worked example: Calisto at Mach 1.5. Calisto’s boattail is 60 mm long from 127 mm to 87 mm: a = 0.469, l/d₁ = 0.472, θ = 18.4°. The chart’s x is √1.25/(2 × 0.472) = 1.18, where it gives C_D = 0.219. A Prandtl–Meyer turn of 18.4° from Mach 1.5 gives C_p = −0.398, so the limit is 0.398 × 0.531 = 0.211, and the attached boattail drags 0.211. At 18.4° it is 17% of the way from 16° to 30°, so the blend takes it 17% toward the base drag on the annulus, 0.167 × 0.531 = 0.088: 0.190, where the rule gave 0.066. For the base, a_b = 0.469: at Mach 2.5 Love’s cylinder gives −C_p = 0.1195, so p_cyl/p = 1 − 0.1195 × 0.7 × 2.5² = 0.477, and p_bt/p = 0.477/(0.442 + 0.558 × 0.469) = 0.678, so k = 0.322/0.523 = 0.616; after the blend 0.683, so the base’s drag falls from 0.078 to 0.053. Calisto’s C_D0 at Mach 1.5 rises from 0.443 to 0.542. The module’s documentation test runs this example (hpr_aero::afterbody).

Against measurements (drag-vs-mach.json, sections boattails, base_pressures and second_order_theory, from the readings in measured-boattails.json; tests::boattails_against_measurements). Every source is a wind tunnel with a turbulent boundary layer and the jet off; each value was read from the report’s figure, with its reading uncertainty in the file. The last column says whether the rows helped build the model: those check it on its own data.

The 0.0102 and 0.0123 are pins on these readings, not tolerances. Below Mach 0.8 Niskanen’s rule, which gives nothing to a boattail longer than three times its drop in diameter, gives Compton’s and Cubbage’s long, gentle boattails 0 to 0.009 where they measure 0.011 to 0.075 (issue #73).

What it leaves out.

• Steep boattails in a thick boundary layer read high. hpr reads Cubbage’s 16° boattails, in a boundary layer a fifth of the diameter thick, 26% to 54% above the measurements, though the flow is still attached. The Arcas Robin’s 15° boattail sits in one thicker than its own drop in radius, and its forebody reads +13.5% to +24.1% from Mach 1.5 (below). No source here gives a correction, so none is applied (issue #72); MIL-HDBK-762 advises boattails under 8° to avoid separation ([762] p. 5-12).
• Through Mach 1 it reads low for gentle boattails: the straight line from Mach 0.8 misses the measured rise’s later, steeper climb and its peak.
• Separation’s 16° and 30° come from one report at Mach 0.6 to 1.28. Between 10° and 30° no attached boattail was measured faster than Mach 1.28, so a boattail of 12° to 20°, like Calisto’s 18.4°, rests on the least-validated part of the model, and likely reads high.
• For one length and area ratio, Jack found the cone’s wave drag the smallest of three shapes ([J53] p. 1), so a curved boattail likely drags more than hpr gives. The chart’s 0.70 and 0.80 curves read up to 32% above Jack past x ≈ 1, 0.0084 at most.
• A tube behind a boattail loses the base’s relief over one drop in diameter, and the 3°, 10°, quarter and half that shape the merge and the wake are judgements, with no measurement behind them but the Arcas Robin’s lip. Its own 1.3 mm length is a gap too: the base keeps 0.944 of its relief.
• A straight cone drawn in parts drags as one cone. A part’s share can be below 0, where the chart makes the longer cone drag less than the shorter: extending the boattail lowers its drag. The one exception: behind a boattail it only partly merges with (a turn of 3° to 10°), a cone’s parts may drag a little less than the whole: an 8° cone behind a 14° part reads the same in 2 or 4 parts, up to 0.45% lower in 8, and up to 2.2% lower drawn in hundreds, worst near Mach 1.
• The 1° below which a part merges only in part, and the fade of a step’s corner over one drop in diameter (flow behind a backward-facing step reattaches farther downstream, so this likely understates a lip’s shelter), are judgements too.
• Nothing models the jet. Fig. 5-141 is measured with the motor off, as is the base drag it scales, and under power hpr applies both to what the motors leave of the base.

Drag limits

• The buildup covers Mach 0 to 5 and refuses Mach 5 and faster, like the normal force (drag::BUILDUP_MACH_LIMIT); an override table takes any Mach number.
• It reads high against the one wind tunnel it has been measured against, at every reading: with the fins off, 12% to 54%, most of it the model’s steep boattail; with the fins on, past Mach 1.2, far more, from the fins (Verification). The fins’ leading edge takes [N09]’s rounded-edge formula, a blunt edge’s, for the airfoil and rounded sections alike. Nothing models the wave drag of a thin, sharp fin, which is far smaller: the Arcas Robin’s four double-wedge fins measure 0.046 at Mach 4.63, against hpr’s 0.30.
• Against MIL-HDBK-762’s worked example, fins left out, the body reads high from Mach 0.9 to 1.2 (the nose and the base) and 6% to 10% low from Mach 1.6 (friction and the base) (Drag against MIL-HDBK-762’s sample calculation).
• It reads low against RASAero II’s Calisto past Mach 1.6, −14.9% at Mach 2 (Drag against RASAero II through Mach 2). Part of that is the body’s lean above; plausible fin inputs bring most rows within 10%.
• Through the transonic rise, from Mach 0.8 to 1.2, the measured shapes follow Stoney’s curves, and cones and ogives Niskanen’s closed form, which reads 45% to 105% above Stoney’s measured 3:1 cone there (the cross-check above).
• Shoulders and boattails past Mach 1. A shoulder takes the nose method, which [N09] calls “somewhat dubious at supersonic velocities” (p. 48), except a lip in a boattail’s wake, which loses a share of it. A boattail keeps eq. 3.88 to Mach 0.8, a rule “based primarily on subsonic data” (p. 49), which over-predicts the Arcas Robin’s 15° boattail and gives long boattails nothing (issue #73). Faster than sound its wave drag reads high for steep boattails in a thick boundary layer (Boattails faster than sound).
• Stoney’s curves end at Mach 1.94 to 1.99 (panel (a)) and 3.59 (panel (b)); past that hpr holds their last value, which panel (b) puts within 8% for two shapes (above).
• Stubby noses. Below fineness 1 a cone or ogive blends toward the flat face at every Mach number, so at rest it reads above eq. 3.86: a cone of fineness 0.5 gives 0.547 against eq. 3.86’s 0.400. And two routes to one shape disagree: a tangent ogive and an ellipse of fineness 0.5 are both hemispheres, but the ogive rises from rest by that blend, which no measurement backs, while the ellipse takes Hoerner’s measured 0.01 to Mach 0.8 and then rises in a straight line (blunt ellipsoids). Below Mach 0.8, trust the ellipse’s.
• Before M1.8b1 the buildup held nose and shoulder pressure drag at its value at rest and refused Mach 1.
• Nothing models laminar flow, fin-tip vortices, interference drag, fin tabs, fillets, canted fins or the flow a boattail guides into the base ([N09] p. 51).

Validity and open questions

• A flared body’s supersonic normal force rests on one measured flare. Since M1.8e17 a conical flare flies the shock-expansion method while its corner’s shock is attached, which moves a flared rocket’s center of pressure forward by a tenth to a quarter of a calibre against the model it had before. The one measurement beside it is TN D-4865’s model 2, an 18.5° flare on a 2.75° cone: −1.9% and +7.0% at Mach 1.9 and 2.3, +13.4% at 2.96, and +51.5% and +50.4% at 3.95 and 4.63, where that flare’s boundary layer is separated (What a marched flare is worth). No other flare angle, and no other body, has been checked. Two more things about that model are open: the attachment test is a wedge’s limit, not a flare’s own (A flare through the method), and a near-flat flare has its one element read by the older generalized method, which leaves a step at the corner’s crossing: +0.129% of the normal force and 0.0051 calibres on the tests’ rocket, and +4.3% and 0.19 calibres on a body whose shoulder is short, where the region sits at degrees rather than thousandths of one (A near-flat flare).
• A step in radius is unmodeled, and nothing measures one faster than sound. Any mismatch of radius at a joint (past 2.7e−11 m between two tubes, or past 1.3e−13 m stepping up at a boattail’s fore end, both far below any tolerance anyone builds to) takes the whole body off the shock-expansion method at every speed. On the tests’ straight rocket that is worth −8.65% of the normal force and 1.03 calibres of center of pressure at the threshold, and −12.55% and 1.36 calibres at a 2 mm step down; on the boattailed one, −11.34% and 1.10 calibres (A step in radius). No published source gives a stepped body’s normal force faster than sound, so the present behavior is not known to be right either; stopping the march at the step was built and rejected (ADR-049, issue #87: a step has no model of its own).
• The body alone misses the 15% target on six of eleven wind-tunnel rows, the one M1.8e set for the body faster than sound, by +37.7% at worst (the short Arcas Robin at Mach 1.5) and within 5% at Mach 3.96 and 4.63. The likeliest cause is Jorgensen’s crossflow term reading too large at the few degrees a slope is fitted over, but the measurement cannot split its own slope from its curvature cleanly, and one row points at the method instead. Closing it needs a cited rule for how the crossflow term grows from zero over the first few degrees, or measurements at finer angles than the reports plot; neither is in hand, so the gap is left visible (The body alone, against the 15% target).
• A center of pressure means little where the body’s normal force is near zero. A deep, steep transition can remove almost all the lift the nose and tube carry, and hpr still divides the moment by what is left: one test shape reports its body’s center of pressure 160 calibres ahead of its own nose tip at Mach 1.2 (issue #104: a near-zero normal force gives a meaningless center of pressure).
• A blunt tip’s handover is capped where the method’s march still settles, not where its theory runs out. The cap is 24° and the cone slopes reach 30°; at 30° the committed Arcas Robin nose reads nearer the report’s own sphere-cone but its answer starts to follow the element count above Mach 4, so the cap stays. What that is worth is measured on both sides in What the cap is worth, and the way out is issue #108: a steeper handover puts the march into η < 0 above Mach 4.
• A boattail steeper than 16° is worth 0.67 to 1.35 calibres of doubt, the most at the lowest supersonic speeds. Nothing measures a separated boattail’s supersonic normal force; hpr holds the measured correlation at 16° rather than letting it fade, which is the conservative end (A steep boattail reads the correlation no steeper than 16°).
• These are small-angle models. α is accepted over [0, π], but fin slopes stay linear in α and nothing models stall. The flight engine uses them at every angle all the same (Rigid-body flight), so its results are least trustworthy where large angles occur: off the rail in a strong crosswind, and near apogee.
• Body lift in wind (measured by flying both codes, not against a real flight; ADR-026, ADR-037). A rocket that leaves the rail slowly in a crosswind meets the air at a steep angle. Juno III, one of RocketPy’s example rockets, leaves at 18 m/s in an 8.5 m/s wind, 26° off the airflow, and there body lift is nearly half its normal force. Much of it acts ahead of the rocket’s center of mass, the nose’s above all, so it moves the center of pressure forward and weakens the moment that turns the rocket into the wind; hpr turns into it less than RocketPy, whose normal force has no body term. Its sideways push alone is about a sixth of body lift’s effect on the drift. Juno III’s apogee ends 245.3 m from the pad in hpr and 396.6 m in RocketPy; body lift is about half of that difference, and hpr’s rail release and fin slope most of the rest. Body lift’s size matters there. Flown in RocketPy with hpr’s model, Juno III’s apogee drift is 248.3 m with Jorgensen’s crossflow (Body lift, η C_dn about 0.91 at that speed), and with a constant K across [G]’s range from 194.1 m at K = 1.5 to 240.2 m at 1.0 (231.1 m at 1.1, hpr’s before M1.8e6); it would be 328.0 m with no body lift. Calisto, off the rail at 28 m/s and 11°, changes its drift by under 0.5% across that range. Which is nearer a real flight is open: the real flights so far compare heights, not drift.
• No airfoils. Fins use the flat-plate lift slope (2π per radian in two dimensions). An airfoil lift curve, such as the one Juno III’s example gives its fins, is not modeled; RocketPy uses it, and its fin slope there is 7.6% steeper (ADR-026).
• In one measured case, fins at α = π/2 give C_N 17.4 against a flat-plate estimate near 5, and at α = π the fins still give 34.7 while every body term vanishes. That case is a 54 mm four-fin rocket at Mach 0.3.
• Speed. The normal force and the drag buildup cover 0 ≤ M < 5; a drag table covers any Mach number.
  • The drag buildup’s transonic and supersonic terms are [N09]’s semi-empirical ones. [N09] expects them “to be reasonably accurate to at least Mach 1.5” (p. 94); against the one wind tunnel they read high from about Mach 1.2 (Drag limits). On the one supersonic flight compared with OpenRocket, hpr’s supersonic pressure drag is about twice OpenRocket’s (#222: which is right is open).
  • The normal force between Mach 0.8 and linear theory’s start M_s is the straight-line join of Fins through Mach 1, which the wind tunnel shows missing by up to +27.1% in slope and 2.36 calibres in CP. Past Mach 3 its body terms read low (Normal force through Mach 1).
  • [N09] eq. 3.35–3.36 would start moving a fin set’s CP aft at Mach 0.5, to about 0.30 of the way along its mean aerodynamic chord (MAC, defined under Fins) at Mach 0.8 for fins of aspect ratio 1.6 (a measure of how long the span is against the chord). hpr keeps 0.25 to Mach 0.8: the Arcas Robin’s measured CP moves forward, not aft, from Mach 0.6 to 0.8.

Verification

• Barrowman’s worked examples (hpr_aero::tests::barrowman_worked_examples). Inputs and printed results, with page numbers, are in the fixture validation/fixtures/aero/barrowman-worked-examples.json. Every printed component and total must agree within 1%. Measured:

  example	C_Nα: hpr / printed	CP: hpr / printed (in)
  Testbed II [B66] pp. 41–45	21.397 / 21.44 (−0.20%)	16.703 / 16.7 (+0.02%)
  Aerobee 350 [B66] pp. 47–50	21.449 / 21.5 (−0.24%)	390.48 / 391 (−0.13%)
  Javelin [TIR] pp. 21–22	35.927 / 35.9 (+0.07%)	11.286 / 11.3 (−0.13%)
  Recruiter [TIR] pp. 23–25, hpr’s model: outside 1%	36.416 / 35.4 (+2.87%)	15.665 / 15.6 (+0.42%)
  Recruiter with TIR-33’s six-fin rule substituted	35.415 / 35.4 (+0.04%)	15.627 / 15.6 (+0.17%)
  Arcon-Hi, two stages [TIR] pp. 27–29	96.163 / 96.2 (−0.04%)	20.803 / 20.8 (+0.02%)
  Arcon-Hi, sustainer alone	32.257 / 32.2 (+0.18%)	17.845 / 17.9 (−0.31%)

  • With hpr’s own model, every CP agrees within 1%, and every slope but the Recruiter’s. Its six-fin slopes are +3.42% (fins) and +2.87% (total). Those are the only 2 of the 38 printed values (19 slopes, 19 CPs) outside 1%, and the test pins that list.
  • With TIR-33’s six-fin rule substituted for the Recruiter, the worst is the Testbed II nose CP, −0.77%: [B66]’s 0.466 L fit against the integrated tangent ogive.
  • CPs are compared as stations from the nose tip. Measured from each part’s own front, two printed values miss 1%: the Testbed II boattail (0.655 in against 0.72 in, −9%, Barrowman’s diameter ratio slip) and the Javelin fins (0.653 in against 0.66 in, −1.1%, rounding).
  • Recruiter’s six fins. TIR-33 scales six fins by N/2 with K = 1 + 0.5 R/(S + R) and no fin-count factor. With hpr’s own rule (0.913 and the full K), the fins are +3.42% and the total +2.87% from the print. The difference between the two rules accounts for +3.22% and +2.83% of that. The test checks the TIR-33 rule within 1% (slopes and CP weighting), reports hpr’s own values, and checks that the rules differ by more than 2%.
  • The printed mid-chord lengths were measured or rounded. hpr computes them from the geometry (Aerobee: 39.7 in printed, 40.50 in geometric). The fixture’s notes list each slip in the printed arithmetic.

• Loft lessons, each with what it concerns:

  • L7, a fin slope and CP that never changed with Mach: fins::tests::fin_cna_compressibility_reduces_to_barrowman_at_m0
  • L8, no correction for five to eight fins: fins::tests::six_fin_cna_applies_fin_count_factor
  • L9, the conical transition’s CP formula used for every shape: body::tests::ogive_transition_cp_uses_volume_form
  • L10, elliptical fins given a trapezoid’s sweep: fins::tests::elliptical_fin_cna_uses_zero_midchord_sweep
  • L89, Barrowman’s values worked by hand, a check kept from Loft’s tests: tests::barrowman_hand_values. A cone’s slope is 2 with its CP at 2L/3 (L its length); a conical transition from 20 to 40 mm radius over 0.1 m is 1.5 at 0.05556 m aft of its fore end; an elliptical fin’s CP is 0.28779 c_r aft of its root leading edge.

• Limits and invariants (body::tests, fins::tests, model::tests):

  • Cylinders and thin transitions; body lift at 0 and 90°.
  • Eq. 57 and 76a closed forms against the same trapezoid as a polygon (1e-13).
  • A 2000-gon ellipse, and a jagged fin.
  • Prandtl–Glauert against [B67] eq. 3-6 written with the aspect ratio, and its M → 1 limit.
  • Roll sums against direct sums; a two-fin set along and across the flow.
  • Mach changes only the fins.
  • Supersonic linear theory on a rectangle, whose slope and CP have closed forms, including a tip cone that crosses the root; outlines of each planform; where linear theory starts.
  • A proptest (a rule checked on many random inputs): scaling every length leaves slopes unchanged and scales the CP; the reference diameter scales slopes only.
  • Refusals: nine fins, Mach 5 for the normal force and Mach 1 for the drag buildup, angles out of range.

Normal force through Mach 1

Two references, in the fixture validation/fixtures/aero/normal-force-vs-mach.json, which cargo xtask aero writes and tests::normal_force_against_mach recomputes and pins (ADR-027). The targets, set before measuring: C_Nα within 15% and the CP within 0.5 calibres (a calibre is one reference diameter). 13 of the 37 rows miss, each for a measured reason below.

• A wind tunnel. NASA tested half-scale models of the Arcas Robin sounding rocket from Mach 0.6 to 4.63 ([D4013], [D4014]): a nose 4.2 calibres long, a cylinder, a 15° boattail and four trapezoidal fins swept 30°, 18.2 calibres long in all, and a longer version of 23.8. The reports print only plots, so their points were read off the pages into validation/fixtures/aero/arcas-robin-wind-tunnel.json, with every figure and page, each C_N to about ±0.01 to ±0.02. The designs model the reports’ nose, a table of coordinates rather than a named shape, as a power-series nose with the same volume, which sets its slender-body CP, and a planform within 2.4%. The slope is the straight line fitted through the plotted C_N from about −4° to +4°, so hpr’s C_N is fitted the same way at the same angles. Its CP is taken over −2° to 2°, the reports’ low angles.
• RASAero II, another code, for Calisto from Mach 0.1 to 2.0: its potential-flow slope (the attached-flow part, without the crossflow lift its export adds from Mach 0.95) and CP from the export RocketPy’s first commit shipped, against hpr’s small-angle values.

The short model (the Arcas Robin itself, 18.2 calibres long), rows outside the targets in bold. The last column is the body alone: the fins-off wind-tunnel reading, and hpr’s body terms fitted the same way. The design’s lip sits in the boattail’s wake and carries nothing faster than sound (A lip in a boattail’s wake), and its vertical tip flies behind a Newtonian cap (Blunt tips), so the body flies the method to its base from Mach 1.2: fins off it reads 3.02 to 3.95 per radian from Mach 1.5, where it read 1.90 to 2.09 on slender-body theory. Below the join its body lift grows as sin² α and with the crossflow Mach number, so their fitted slope moves a little with Mach and with the angles each plot happens to cover (1.90 to 2.09). From Mach 0.6 to 1.2 the fins-off readings, on a coarse grid (±0.02 per point), scatter from 1.41 to 2.88 with no trend, so they don’t settle whether hpr’s 1.91 is high there.

What the misses come from:

• Faster than sound, every slope now passes, and the long model’s CP at Mach 1.8 and 2.3 does not. Until M1.8e7 and M1.8e8 the committed designs’ vertical tip and lip kept the shock-expansion method off, so their bodies flew slender-body theory past Mach 1 and lifted 2.0 to 2.6 per rad where the tunnel’s body alone lifts 2.2 to 4.6: the short model read −16.3% at Mach 2.96 and −28.0% at 4.63. With the cap (Blunt tips) and the lip carrying nothing in the boattail’s wake (A lip in a boattail’s wake), the body grows with Mach, as the measurement does though not as steeply (3.02 to 3.95 per rad fins off on the short model, against the tunnel’s 2.19 to 4.15). The whole rocket’s rows from Mach 1.5 read +8.8% to −3.3% (short) and +9.4% to −2.3% (long). What is left is where the body reads high: fins off it is 15% to 19% above the tunnel at Mach 1.8 and 2.3 on the long model, which pulls the whole rocket’s CP 0.53 and 0.52 calibres forward of the measured one, just outside the half-calibre target. M1.8e6 sized that excess and left it (ADR-037); the body alone is judged against the 15% target in The body alone, against the 15% target, where it is outside on six of eleven rows. The fins’ share (the fins-on reading less the fins-off one) agrees with hpr’s fins within −1.4% to +7.0% at Mach 3.96 and 4.63, with about 5% of doubt of its own: over the boattail the models’ fin roots follow its 15° surface below the cylinder, and the design leaves that strip out, about 0.32 in² of each fin’s 5.8 in² (5.5%).
• Mach 0.6, within the targets by errors that cancel. Both models pass there, but hpr’s body is 25% and 19% above the fins-off readings, which are poorly determined at these speeds, and its fins’ share 9.3% and 3.9% below the measured one.
• Transonic, Mach 0.8 to 1.2. The fins’ measured share lifts less at Mach 0.8 and 0.9 than at 0.6, then jumps at Mach 1. hpr’s fins lift more, by Prandtl–Glauert and then along the join to linear theory’s peak at M_s (1.2 for these fins). The long model’s CP jumps forward at Mach 1, 2.36 calibres from hpr’s. No closed-form method covers this region, and the join is not fitted to it.
• RASAero II keeps its slope and CP constant through subsonic flow, where hpr’s rise with Prandtl–Glauert, so they part from Mach 0.8. Past Mach 1.2 Calisto’s von Kármán nose flies the shock-expansion method behind a Newtonian cap, so its cylinder carries lift: Mach 1.5 reads +12.9% and Mach 2 +8.3% (−3.1% and −16.8% on slender-body theory, before M1.8e7). The wind tunnel sides with neither there. Below that the agreement is partly by construction: the Calisto design has the 2018 fins because they reproduce this export at low speed (ADR-009). Past Mach 1 the result rests on the choice of RASAero’s columns: against its secant slope and CP to 4°, which include its crossflow lift, 3 of the 11 rows from Mach 0.8 are within the targets (the tightest, Mach 1 by 0.00002 calibres), not 7, and Mach 2 is −9.6%. That comparison is a summary in the fixture’s secant_comparison, not a second set of rows: the export stays in refs/, and the fixture commits its values once (ADR-009, ADR-027). Calisto has no fins-off data, so its body and fins can’t be split as the wind tunnel’s can.

So, for fins like these, whose linear theory starts at M_s = 1.2: from Mach 1.5 up, trust hpr’s slope to about 10% (the rows run +9.4% to −3.3%) and its CP to about half a calibre, which the long model misses by 0.03 at Mach 1.8 and 0.02 at 2.3; between Mach 0.8 and M_s, in the join, neither. A fin set’s own M_s is FinSetAero::fin.supersonic_mach, from AeroModel::fin_sets. Fins swept further back start later: a leading edge swept 48° starts at Mach 1.5, and until then it is in the join. Nothing past Mach 4.63 has been checked, though the model runs to 5.

The body alone, against the 15% target

What this covers: how far hpr’s body alone is from NASA’s measurement of the same body, and where what is left of the gap sits. How far to trust it: at Mach 3.96 and 4.63 the two agree within 5%; below that hpr reads up to 38% high. On five of the six rows outside the target most of that excess is body lift; on the sixth it is the method itself.

The milestone M1.8e set a target before any of this was built: the Arcas Robin’s body alone within 15% at every Mach number from 1.5, and both configurations’ whole-rocket slope within 15% at Mach 3.96 and 4.63. The second half is met (+2.8% to −3.3%). The first is not, on six of eleven rows, and this is where they stand (ADR-040). Reading the table:

• Rows outside the target are in bold. Slopes are per radian on the body’s cross-section.
• M/f_n is the Mach number over the nose’s fineness, the argument TN 3527 ([SD56]) states its method for from 0.4 to 2. One row, the short model at Mach 1.5, is below that at 0.36.
• measured and hpr are the straight-line slopes fitted at the tunnel’s plotted angles, as ADR-036, which fixes how these comparisons are fitted judges them.
• at α → 0 is the slope at zero angle. hpr’s is the method alone, since body lift vanishes there; the measurement’s comes from fitting its points with C_N = a α + b α |α|, the form the tunnel’s own curves follow, and a is quoted with its standard error.
• curvature is the rest: the fitted slope less the slope at α → 0. For hpr it is body lift.

What the rows say, and what they can’t. What is solid is the first three number columns: on six rows hpr’s fitted slope is 15% to 38% above the tunnel’s, and at Mach 3.96 and 4.63 it is within 5%. The split into a slope at α → 0 and a curvature is softer, and it is worth saying why before leaning on it. The tunnel plots seven points over about ±4.5°, and in a fit of C_N = a α + b α |α| over so short a span the two terms trade off almost exactly: their correlation is −0.96. A fit that reads a low must read b high. So the measurement’s own split carries the standard errors in the table (±0.30 to ±0.42 per radian, the widest of them on a slope of 1.920), and the curvature, being the same slope subtracted from another, carries at least as much.

Where the gap most likely is. With that said: at α → 0 hpr is within 1.5 standard errors of the measurement on every row outside the target (0.2 to 1.5 of one), so the readings cannot convict the shock-expansion method, the Newtonian cap or the boattail’s measured share. On five of those six rows most of the fitted gap sits in the curvature instead: what the rest of the plotted angles add, which for hpr is body lift. The sixth is the short model at Mach 2.96, where 77% of the gap is hpr’s slope at α → 0, 1.2 times the measured one: there the method itself, not body lift, carries most of the miss. Hpr’s curvature is 1.3 to 2.8 times the measured one below Mach 2.96, 1.1 to 1.3 times it at Mach 2.96, and 0.89 to 1.27 times it at Mach 3.96 and 4.63. The ×12.20 on the short model at Mach 1.8 is not a measurement of anything: the tunnel’s own curve barely bends there (0.094 per radian, against an uncertainty three times its size), so the ratio’s denominator is consistent with zero.

The likeliest single cause is Jorgensen’s crossflow term, which ADR-037 chose because the tunnel’s high-angle points support its size: at the few degrees these slopes are fitted over it reads too large. Two other explanations are open and the readings do not close them: the short model at Mach 1.5, the worst row at +37.7%, sits at M/f_n 0.36, below the 0.4 that TN 3527 states its method for, and the same model at Mach 2.96 points at the method rather than at body lift.

What would close it. A cited rule for how the crossflow term grows from zero over the first few degrees, or measurements of this body at finer angles than the reports plot. Neither is in hand, so the gap is left visible here rather than tuned away. The rows are in validation/fixtures/aero/arcas-robin-body-gap.json, written by cargo xtask aero, and a test pins which rows are outside.

Checking the shock-expansion method

The second-order shock-expansion method of Bodies faster than sound, which a flight uses from Mach 1.2 on the bodies it covers, against two references in the fixture validation/fixtures/aero/shock-expansion.json. cargo xtask aero writes it, and shock_expansion::tests::against_tn3527_and_the_arcas_robin recomputes every value and pins every miss (ADR-033).

To check a share by hand from outside the crate, ShockExpansionBody::element_flows reports each element’s flow (the state behind its corner, the tangent cone it relaxes toward, how fast it does so, and the radius eq. 19 needs), which is what the library’s own hand integral of a boattail and the tube behind it uses (footnote_eights_boattail_share_by_hand).

The tip cone’s flow (cone_flow_agrees_with_naca_1135_charts), against [R1135]’s cone charts 5 to 7 at Mach 1.5 to 3 and cones of 10° and 20°: the shock angle within 0.3°, the surface pressure coefficient within 0.004 and the surface Mach number within 0.015, twice the charts’ reading error, since they are drawn for γ = 1.405 and hpr uses 1.4. At Mach 2 on a 10° cone hpr gives a shock at 31.21° against the chart’s 31.25°.

The report’s own tables ([SD56] Tables I and II, transcribed from the page images into tn3527-bodies.json). They cover 144 bodies: cones and tangent ogives of fineness 3, 5 and 7, on cylinders of 0 to 10 calibres, at Mach 3, 4.24, 5.05 and 6.28. For each they give the method’s slope and CP, as its authors computed them by hand in 1956, and, for all but the 8-calibre cylinders, NASA’s wind-tunnel measurements. The targets were set before measuring: within 0.05 per radian and 0.1 calibres of the report’s values, and within the ±0.2 per radian and ±0.2 calibres the report claims against its measurements. Each cell counts the rows within, with the range of hpr’s value less the reference’s:

The 12 cones with no cylinder only read Fig. 2 back, so against the report’s values they check the hand reading, not the method.

A worked example: a cone of fineness 5 on a cylinder 4 calibres long, at Mach 4.24. Slender-body theory gives 2 per radian at any length. The tip cone alone gives 1.868 (Fig. 2). With the cylinder hpr gives 2.922, the report 2.91, and the wind tunnel 2.84.

In all, against the measurements, 117 of 120 slopes and 109 of 120 centers of pressure are within the report’s ±0.2; against the report’s own values, 102 of 144 slopes and 125 of 144 centers of pressure are within 0.05 and 0.1. That is 75 of the 528 comparisons outside, so the targets are not met (ADR-033 records it):

• Against the report’s own values (61 misses), in two kinds.
  • Where the march stays inside the method’s limit (49 misses), hpr follows the printed equations and the printed values depart from them. The largest are the fineness-7 cone on long cylinders (hpr high, up to +0.146 at Mach 6.28 over 10 calibres) and the ogives (hpr low, most at fineness 5 and Mach 3, down to −0.134). The evidence is a second implementation of the same equations, written from the paper during this work with its own cone solver. For the cone-cylinders it used the report’s closed form (its Appendix C), for the ogives its ten-element march. With hpr’s hand-read Fig. 2, it agrees with hpr within 0.0001 per radian on all 72 cone-cylinders and within 0.0006 per radian and 0.0003 calibres on the 60 ogive-cylinders that stay inside the limit. It is an uncommitted scratch script by the same author, so it can’t be rerun from the repository, and it can’t catch a misreading both share. Why the printed values differ is not known. The report took its cone pressures from charts (its Fig. 1), and a thin cone’s small pressure differences are sensitive to them; that is a guess, not a finding.
  • Where the march reaches the method’s limit (12 misses), on the fineness-3 ogive at Mach 5.05 and 6.28, near the tip. There hpr’s CP at Mach 5.05 sits 0.19 to 0.67 calibres ahead of the report’s, and the report’s measurements agree with the report, so this is hpr’s gap, not the report’s. The report doesn’t say how it continued past its limit. Carrying the pressure gradient on through the reduced elements comes closer at Mach 5.05 but further at Mach 6.28, and it doesn’t settle as elements are added. This is open (issue #81).
• Against the measurements (14 misses), in four groups:
  • The fineness-7 cone on long cylinders (6). The report is already 0.07 to 0.15 high there, and hpr, following the closed form, adds 0.06 to 0.19 more.
  • Rows where the report is itself 0.20 to 0.22 off (3): the fineness-5 and fineness-3 cones’ CP at Mach 6.28 over 10 calibres, and the fineness-5 ogive’s at Mach 5.05 over 10.
  • The fineness-3 ogive at Mach 5.05 (4): the limit above; the worst misses, −0.278 per radian and −0.540 calibres.
  • The fineness-7 ogive’s CP at Mach 5.05 over 4 calibres (1): −0.206, just outside, where the report reads −0.13.

The Arcas Robin ([D4014], the wind tunnel of Normal force through Mach 1). The method needed a pointed tip when this comparison was made (the committed nose, behind the cap it now takes, is compared in Blunt tips), so here the nose is the secant ogive (a circular arc meeting the body at an angle) through the tip and base that best fits the report’s coordinate table, 4.17 calibres long, so the report’s Mach-over-fineness range of 0.4 to 2 covers Mach 1.67 to 8.3: its arc radius is 1.744 times a tangent ogive’s, it misses the table by 0.003 in rms, and its tip half-angle is 10.76°. hpr’s committed design keeps its power-series nose, whose tip is blunt. The measured slope is the fins-off reading fitted over the plotted angles, as above. It includes the boattail, the lip behind it, and crossflow lift at those angles. The method has neither the lip nor crossflow at α → 0, and takes the boattail only by the report’s footnote 8, so the target is not applied here; it is applied to the body a flight flies, in The body alone, against the 15% target. This table is the method’s own; the body a flight flies since M1.8e6, with the boattail’s measured share, is compared below.

The method’s slope grows with Mach number, as the measurement does: 2.55 to 3.37 on the short model, where slender-body theory keeps its nose at 2. By the end of either cylinder the lift has died away, so the long model gets almost nothing more (3.313 against 3.300 at Mach 3.96), while its measurement is 0.57 higher. That difference goes with the longer body’s larger side area, the mark of crossflow lift. The boattail column here is the report’s footnote 8, the method’s own rule, which a flight used from M1.8e4; since M1.8e6 a flight takes Washington and Pettis’s measured share instead (the worked example under The body faster than sound in a flight).

The table above is the method alone, at α → 0. A flight also adds body lift, which grows as sin² α; the table leaves it out, and the measured line includes it.

Most of that gap was the comparison, not missing lift (M1.8e5 sized each cause of it on hpr’s body model before M1.8e6, Galejs’s body lift and footnote 8’s boattail; the research note has the tables). The measurement is a straight line through points from about −5° to +4°, and crossflow lift, which grows as α |α|, steepens it. Fitted the same way at the same angles, with the body lift a flight added, hpr’s body with its boattail read 14.9% to 73.2% high at every Mach number (fixture arcas-robin-gap.json, which cargo xtask aero writes and aero_gap::tests::committed_fixture_is_current keeps current).

• Crossflow lift steepens that fitted line. In hpr’s body lift it adds 1.40 to 2.09 per radian; in the tunnel’s own points (fitted with an α |α| term, the curvature) it adds 0.09 to 1.74. From Mach 2.3 the tunnel’s curvature matches body lift with a factor K (Bodies of revolution) from 0.66 to 1.05, each ±0.18 to ±0.23 (one standard error), where hpr used 1.1. Jorgensen’s crossflow method ([J77] eq. 2.12, Fig. 4 and p. 15) gives about 0.9 for these bodies at small angles.
• The fit’s slope at α → 0 and its curvature move together (their errors correlate at −0.95 to −0.96), so the readings can’t split hpr’s excess between body lift and its slope at α → 0. With body lift at Jorgensen’s size alone, hpr still reads 8.2% to 60.7% high.
• The one sized cause that size is the boattail’s share: TN 3527’s footnote 8, which a flight used, gives −0.177 to −0.026, slender-body theory −1.324. The lip, which the method can’t take, adds +0.178 by slender-body theory; a blunt tip like the tunnel’s, scaled from a blunter one measured, loses 0.015 to 0.07 past Mach 3. Below Mach 3, where the tangent cones’ slopes are held at TN 3527’s Fig. 2 Mach 3 curve, Sims’s tables ([S64] Table 2, p. 20) move the nose and cylinder’s share by −0.031 to +0.056 at most. hpr’s reading of issue #81 (how it continues the method where TN 3527’s relaxation condition fails) changes nothing on this body.

From then on the Arcas Robin is judged like for like, at the tunnel’s angles (ADR-036).

Crossflow’s size and the boattail’s share, together. Since M1.8e6 body lift takes Jorgensen’s crossflow (Body lift) and a boattail Washington and Pettis’s measured share (The body faster than sound in a flight) (ADR-037). Fitted like for like, as the tunnel’s line is, the same body reads +3.4% to +41.0%. Each change takes about half of the old excess off; from Mach 3.96 both models are within 15%, and from Mach 1.5 to 2.96 hpr still reads 16% to 41% high. The columns are hpr’s body model before M1.8e6, each change alone, and both; the last column is the current model’s slope at α → 0, which leaves body lift out. Slopes per radian on the body’s cross-section; the measured line’s standard error takes each reading’s accuracy as independent.

The rows are in arcas-robin-crossflow.json, which cargo xtask aero writes and aero_crossflow::tests::committed_fixture_is_current keeps current; aero_crossflow::tests::the_guide_quotes_the_fixture checks this table against it cell by cell. What is left at Mach 1.5 to 2.96 can’t be split between body lift and the slope at α → 0 from these readings, as M1.8e5 found. On the short model at Mach 1.5 and 1.8 the tunnel’s points from −5° to +4° barely curve (a factor of 0.32 ± 0.24 and 0.07 ± 0.25 on body lift, where hpr uses about 0.9), while hpr’s slope at α → 0 (1.93 and 2.17) lies within about one standard error of the tunnel’s (1.78 ± 0.32, and 2.52 ± 0.33, which hpr is 1.1 below): there most of the excess is body lift, which at 6° is already about half of hpr’s normal force, and the points at 6° need a factor of 0.47 and 0.58 on it. From Mach 2.3 the curvature gives factors of 0.66 to 1.05, each within about one standard error (±0.18 to ±0.23) of Jorgensen’s, five of the eight below it. The fixture also holds each of the 62 points above +4° and the boattail’s share at α → 0 under each rule. One caution on the long model: its points from −5° to +4° are M1.8a’s reading, which may carry a skew of the page that puts its slope 3% to 5% high (issue #97); its rows here depend on how that is settled.

Where the body’s lift acts. A body model can match the slope with its lift in the wrong place, so the body’s center of pressure is checked too, from the tunnel’s fins-off pitching moment (read for this milestone into arcas-robin-fins-off-moment.json, ±0.02 to ±0.025 in C_m). Both are taken the way the tunnel’s are: straight lines through the pitching moment and the normal force at the plotted angles up to ±4.5°, calibres from the nose tip; the measured ones carry about ±0.5 calibres from the readings. The measured boattail share, which takes lift off at the tail, moves hpr’s center of pressure forward, toward the tunnel’s: before M1.8e6 hpr put it 0.90 to 3.85 calibres aft of the tunnel’s on the short model and 0.59 to 1.94 on the long; now −0.19 to +1.59 and −0.88 to −0.18. Two rows still miss by more than the readings’ half a calibre: the short model at Mach 1.5 and 1.8, where the slope misses most too.

Drag verification

• RocketPy’s drag curves at Mach 0.3 (tests::rocketpy_drag_curves_at_mach_0_3, fixture validation/fixtures/aero/rocketpy-drag-curves.json, written by cargo xtask aero from refs/rocketpy). Every RocketPy example whose curve is labelled RASAero, at sea level in the 1976 standard atmosphere (USSA76; RASAero II computes its exports’ Reynolds numbers there); tolerance 10%. The fixture holds only derived numbers: each curve’s value at Mach 0.3, hpr’s C_D0 and the error. The test recomputes hpr’s C_D0 and the errors from the committed designs, and cargo test -p xtask reruns the comparison when refs/rocketpy is present.

  case	curve	hpr C_D0	error	range over inputs
  Calisto, 2018 fins	RASAero II export, power-off	0.3982	+4.4%	−14.0% to +12.8%
  Calisto, getting-started fins (variant)	the same	0.3537	−7.3%	−12.9% to +19.3%
  Juno III	labelled RASAero II, 3-decimal table	0.3525	−6.0%	−10.5% to +24.1%
  Cavour, power-off	labelled RASAero II, 3-decimal table	0.5034	−8.3%	−22.3% to −0.4%
  Cavour, power-on (outside 10%)	the same, power-on	0.4487	−18.3%	−32.2% to −10.3%
  Valetudo, power-off (outside 10%)	labelled RASAero, 3-decimal table	0.5566	−47.0%	−59.4% to −42.5%
  Valetudo, power-on (outside 10%)	the same, power-on	0.5189	−50.4%	−62.8% to −45.9%

  • Inputs (ADR-009, the drag decision). The exports record none, so the designs follow one declared rule:
    • RASAero II’s default smooth finish.
    • A NACA 00xx airfoil file in the example gives an airfoil section that thick at the mean aerodynamic chord (Calisto’s getting-started fins).
    • A published section is used: Juno III’s team placed second for a technical award, cited for “análise de aletas com perfil de aerofólio truncado” (an analysis of truncated-airfoil fins); taken as rounded at the placeholder thickness, since the citation gives no thickness.
    • Otherwise the placeholder, square 3 mm (Calisto’s 2018 fins, Cavour, Valetudo).
    • Rail buttons are as RocketPy defines them (without them, Calisto is +1.8%).
  • Sensitivity. The range is over square, rounded and airfoil fins (3 mm, or 12% for the airfoil), 0 or 20 µm, and with or without rail buttons. Before the published-section rule, square fins gave Juno III +14.6% and the getting-started Calisto +10.7%. The check places hpr near RASAero’s subsonic drag under a declared rule; without the inputs it can’t show agreement to 10%.
  • Power-on. Separate power-on curves are compared (Cavour’s and Valetudo’s). Subtracting the motor’s area ([N09] pp. 50–51) removes 42% of Cavour’s base drag and 29% of Valetudo’s at Mach 0.3. Cavour’s power-on table is within its 0.001 rounding of power-off from Mach 0.16 up (0.0001 at 0.3) and 0.001 to 0.013 lower below; Valetudo’s is 0.004 lower, about a ninth of hpr’s relief. The designs’ motor diameter is the larger of the grain and nozzle exit diameters, since RocketPy gives no case, and Cavour’s result depends on it: −8.3% with no relief, −14.8% at 54 mm, −18.3% at the design’s 67 mm nozzle exit, −20.8% with the example’s 75 mm motor. The cause of that miss stays open: Niskanen’s rule of taking the motor’s area off the base, a RASAero run with little or no nozzle exit diameter, or tables sampled along a flight (their uneven Mach spacing suggests it; unconfirmed).
  • Valetudo. Its table (1.05) is 1.44 times the OpenRocket export for the same rocket (0.728). With that file’s own inputs (60 µm, two 14 mm × 30 mm lugs, 3 mm square fins), hpr gives 0.714, 1.9% under the OpenRocket export and 32% under the table. As designed for this comparison, its 0.5566 is 23.5% under the export.
  • Not compared.
    • Calisto’s power-on curve, which equals its power-off curve (no nozzle exit diameter in RASAero).
    • Juno III’s power-on drag, which RocketPy takes from the same file.
    • Calisto’s power-on result would be −5.0% (its power-on file is its power-off file).
    • The other examples, whose drag is a constant, CFD or of unknown origin.

• Loft lessons, each with what it concerns:

  • L11, drag that changed with the order of the fin sets: drag::tests::drag_invariant_to_fin_set_order
  • L12, uncited form-factor, friction and roughness constants: drag::tests::form_factor_and_roughness_match_cited_values
  • L13, no relief of base drag while a motor burns: drag::tests::power_on_base_drag_subtracts_thrusting_motor_area
  • L14, uncited launch-lug drag: drag::tests::launch_lug_drag_matches_cited_hollow_tube_formula
  • L15, shoulder drag that jumped as its length went to zero: drag::tests::shoulder_drag_continuous_as_transition_length_tends_to_zero
  • L16, a silent cap on the drag coefficient that hid bad geometry: drag::tests::malformed_geometry_is_an_error_not_a_clamped_cd
  • L90, drag invariants, a check kept from Loft’s tests (skin friction’s published jumps, split fin sets, base drag at Mach 1): drag::tests::skin_friction_follows_eq_3_81_and_drag_invariants_hold

• Limits of every term (drag::tests): friction below 1e4, at R_crit and to Mach 5; stagnation pressure against the isentropic series and its limits either side of Mach 1; base drag at rest, at Mach 1 and far above; the joint term from smooth to a step; the boattail factor’s three pieces and their joins; fin pressure drag by cross-section with the leading edge’s joins at Mach 0.9 and 1 and the sweep; the angle-of-attack factor’s stated values and zero slopes, monotonicity and its sign-reversed mirror (and C_A < 0 at 135°); joint angles of cones and power-series, Haack and ogive noses; curved boattails ending in blunt tips; a tail closing to a point; leading-edge sweeps of trapezoids, kinked outlines and ellipses (against a quadrature to 1e-8); a whole rocket’s buildup written out by hand to 1e-12, its component sum, and override tables (rescaled to another reference diameter).

• Drag through Mach 1 (nose_drag::tests, drag::tests): Loft lesson L17, Loft’s fin leading-edge drag frozen at its Mach 1 value and nose drag with no Mach term, is drag::tests::leading_edge_and_cone_pressure_drag_have_supersonic_branches. A 3:1 cone by hand from rest through eq. B.4, with its joins smooth to 1e-5; the ogive factor at its ends and middle; eq. B.9 through both its anchors; eq. 3.87 meeting its lower bound and its fallback; a step from 0.8 to the flat face; short cones tending to the step and meeting the closed form at fineness 1 (Loft lesson L15); Stoney’s curves reproduced at fineness 3, held past their ends and interpolated between shapes; the guide’s worked example; the 3:1 cone against Stoney’s measured one; refused shapes; and a property test that every shape at any fineness, joint angle and Mach number to 5 gives a finite, non-negative coefficient with no jump at M_L.

• Boattails faster than sound (afterbody::tests): the Prandtl–Meyer function against NACA Report 1135’s table and its limit of 130.45°, and its inverse; an expansion’s pressure by hand and at the vacuum limit; the chart giving back its readings, falling with x and the area ratio, and reaching 0 at a = 1; Calisto’s boattail by hand (the guide’s worked example) with the joins at Mach 0.9, 1 and 1.2; the drag past the chart continuous at its end and closing on the 2D limit; separation from 16° to 30°; the base-pressure ratio on Fig. 5-141’s line; and every boattail from 1° to 89° finite and non-negative to Mach 5. The comparison with measured boattails and Jack’s theory is tests::boattails_against_measurements.

• Boattails in parts, wakes and gaps (drag::tests): a boattail split in two drags as one (a_boattail_split_in_two_drags_as_one), and so does a straight cone of 0.5° to 7° in 2, 4 or 8 parts, where a part’s share is below 0 too (a_straight_cone_in_parts_is_one_cone); a corner keeps two parts apart and the merge is continuous in the turn (a_sharp_corner_keeps_its_boattails_apart); a pair drags between its two limits (soft_merges_stay_between_their_limits); a partial merge shares the flow (a_partial_merge_shares_the_flow); a change of ε in any radius or length behind 2° to 14° boattails moves the drag in proportion to ε (a_part_narrowing_by_nothing_is_a_tube_and_one_of_no_length_a_step); a lip’s wake by its rise and its gaps (a_lip_in_a_boattails_wake_fades_with_its_rise, a_lip_drawn_as_a_step_up_is_a_lip, a_hairline_step_before_a_lip_changes_nothing); a retainer behind a step down (a_retainer_behind_a_step_down_is_in_its_wake); and a 40-part zigzag keeps its tails few (a_zigzag_boattail_keeps_its_tails_few).

• Tables (table::tests): RocketPy’s quirks (\r\n, 01.05, a repeated row), a byte-order mark, quoted fields and trailing commas, RASAero II’s header with rows at 2° and 4° skipped, and malformed text (a bad first row, repeated or unsorted Mach numbers, nan) by line.

Drag against the Arcas Robin wind tunnel

NASA measured the axial force on its half-scale Arcas Robin models, the same ones as the normal force above, from Mach 0.6 to 4.63 ([D4013], [D4014]), fins at 0° and with the fins off. The models sat on a sting, so their base pressure isn’t a free flight’s, and both reports take the base apart: [D4013] plots the axial force “corrected for base axial force” (C_A,corr, Figs. 11–12), and [D4014] the axial force and, separately, the force on the balance chamber inside the base (C_A,c, Figs. 4–6). What compares, then, is the forebody: hpr’s C_D0 less its base drag (friction, pressure and parasitic drag) against the measured axial force with the base at the free stream’s pressure. For [D4014] that is C_A − 1.383 C_A,c, which takes the chamber’s pressure over the whole base as [D4013]’s correction does: 1.383 is (1.470/1.250)², the base’s diameter in inches over the 1.250-inch cavity drawn in [D4014] Fig. 1(a), squared. The report states no chamber area, and taking it over the chamber alone moves the measured values by 0.002 to 0.015, which changes no row’s verdict. The readings are in arcas-robin-wind-tunnel.json, read off the reports’ plots with their figure, page and reading uncertainty (±0.002 in C_A for most, against the reports’ own ±0.004).

hpr flies the committed designs (Normal force through Mach 1) at both tunnels’ Reynolds number, 3.0 million per foot, with two inputs set for drag before measuring: the double-wedge fins take hpr’s airfoil section, as Niskanen modeled them ([N09] p. 90), and the machined steel models a polished finish, 0.5 µm, since the reports state none. The target, set before measuring, was M1.8’s 10% for drag. cargo xtask aero writes drag-vs-mach.json, each row with hpr’s drag by part, and tests::drag_against_mach recomputes it from the designs and pins the 2 rows of 44 within target. The two input choices matter, and moved hpr toward the tunnel: with square edges and the default 20 µm finish no row is within target, with the airfoil section alone none, with the polished finish alone none, and with both 2 (tests::drag_against_mach_depends_on_the_fins_and_finish). The airfoil section follows the drawings and Niskanen; the finish is a guess. Allowing each reading its uncertainty and the reports’ ±0.004, neither of the 2 could fall the other side of 10%. Before M1.8b3 modeled the boattail faster than sound and the lip in its wake, 8 rows were within target, 6 of them because the lip’s 0.085 made up for the missing wave drag. Forebody drag on the reference area, measured and hpr’s, and hpr’s error:

Why it misses, from the drag by part in the fixture:

• The fins past Mach 1.2. hpr’s fins add about 0.30 from Mach 1.5 up, where the measured fins-on less fins-off falls from 0.153 at Mach 1.5 to 0.046 at 4.63: +78% at Mach 1.5 and +551% at 4.63 on the short model. The leading edge takes [N09]’s rounded-edge formula, whose value grows toward 1.2 on the fins’ frontal area, where a thin, sharp fin’s wave drag falls with Mach. Niskanen’s own comparison with this wind tunnel shows the same, his simulation about 80% high by Mach 3.96 ([N09] Fig. 6.6, p. 90). At Mach 0.6 hpr’s fins are +10% and −15% of the measured increment; from 0.8 to 0.9 they rise sooner than the measured fins do.
• The boattail faster than sound. hpr’s 15° boattail drags 0.285 from Mach 1.0 to 1.2, 0.196 at 1.5 and 0.037 at 4.63 (Boattails faster than sound). If the rest of hpr’s forebody were right, the tunnel’s boattail would drag about 0.12 at Mach 1.5 and 0.08 to 0.11 at 1.8, 40% to 94% under hpr, and next to nothing from Mach 3.96. Cubbage’s 16° boattails, in a boundary layer a fifth of the diameter thick, read high the same way, 26% to 54%; NASA reports the flow separating over this boattail at the higher Mach numbers ([D4014] p. 6). With the fins off hpr reads +13.5% to +24.1% from Mach 1.5 (issue #72).
• The lip. The models end in a lip 1.3 mm long that flares from the boattail’s 33.2 mm to the base’s 37.3 mm. It sits in the boattail’s wake, and hpr gives it no pressure drag. Before M1.8b3 hpr took it as a shoulder in the free stream, a stubby cone worth 0.065 at Mach 0.6 and 0.084 to 0.086 from Mach 1.2, which made up for the missing wave drag and put six rows within 10%. The short model also keeps the fins’ raised root fairings with its fins off, which hpr leaves out and TN D-4013 blames for its higher drag from Mach 0.975 to 1.2 (pp. 4–5).
• The boattail below Mach 1. [N09]’s boattail rule (eq. 3.88) gives the 15° boattail a pressure drag of 0.063 at Mach 0.6 (its 0.070 less its friction). The tunnel’s forebody holds that same pressure on the boattail’s surface, and on the short model the whole forebody with its fins off measures 0.22 there, against hpr’s friction alone of 0.19: little is left for the boattail’s pressure. The rule over-predicts this boattail, as Niskanen found against the same tunnel ([N09] p. 90). At Mach 0.6 and 0.8 hpr’s forebody with its fins off is +12.0% to +22.8% high, most of it the boattail rule.
• Through Mach 1, where drag rises steeply, the measured forebody with fins off jumps from 0.29 to 0.42 between Mach 0.95 and 1.0 on the short model. hpr’s boattail rises from 0.076 at Mach 0.8 to 0.285 at 1.0, sooner than the tunnel’s, and the forebody with its fins off reads +38.8% to +54.1% from Mach 0.9 to 0.95 and +20.3% to +50.8% from Mach 1.0 to 1.2.

What this shows: hpr’s drag reads high for this rocket at every Mach number, from about Mach 1.2 most of all by its thin, sharp fins, which hpr takes as blunt, and at every speed by its steep boattail. The body alone reads +12.0% to +54.1%; before hpr modeled the boattail faster than sound, with the lip in it and no wave drag, it read −9.2% to +71.1%. hpr’s base drag, which the tunnel can’t measure, is compared with a calculation (Drag against MIL-HDBK-762’s sample calculation) and, behind boattails, with measured bases (Boattails faster than sound).

Drag against RASAero II through Mach 2

M1.8 asks for drag within 10% of the curves labelled RASAero in RocketPy’s example rockets from Mach 0.1 to 2.0, with the errors by band. hpr doesn’t meet that. This section gives the errors, what the boattail’s wave drag changed, and how far the curves’ unrecorded inputs reach (ADR-029, the decision on this comparison, and ADR-030).

How it is compared. cargo xtask aero compares hpr’s zero-lift drag coefficient, C_D0, with each curve every 0.05 from Mach 0.1 to 2.0, wherever the curve reaches. Each point is at sea level in the 1976 standard atmosphere, at the Reynolds number for its Mach number, as RASAero II computes its exports. The designs and their inputs are those of the Mach 0.3 check above. Bands are Niskanen’s (Table 3.1): subsonic to Mach 0.8, transonic below 1.2, supersonic from 1.2. The fixture, rocketpy-drag-curves.json, holds hpr’s value and the error at every Mach number and each band’s summary. It doesn’t hold the curves, but the two numbers give a curve’s value back at each Mach number. drag::tests::supersonic_cd_against_rasaero_tables recomputes every row and pins the counts (Loft lesson L18: Loft, the earlier simulator this project learns from, used an invented transonic drag curve).

The curves don’t all reach Mach 2. Calisto’s, the one real RASAero II export, does. Juno III’s is hand-edited past Mach 0.92: it climbs a constant step per row to Mach 1.0 and then drops to 0.001, so the comparison stops at 0.92. Cavour’s stop below Mach 0.93, and Valetudo’s at 1.53.

Rows within 10%, and the range of the errors, by band:

The getting-started fins are a variant: RocketPy’s getting-started example gives Calisto larger fins with a thick NACA 0012 airfoil, but the export was made for the 2018 fins, whose normal force it matches. So the variant’s rows show how much the fins move drag, not a second agreement: its thick fins now read 23% to 32% high faster than sound. Calisto on its 2018 fins is within 10% up to Mach 0.8, at 0.95 and 1.0 and from 1.2 to 1.55; it reads +13.0% and +16.4% at Mach 0.85 and 0.9, where its boattail’s rise starts sooner than RASAero II’s, −10.1% at Mach 1.05, and falls below the curve from Mach 1.6, to −14.9% at 2.0. Valetudo’s curve is 1.44 times its own OpenRocket export, as the Mach 0.3 check found, and Cavour’s power-on miss is the same open question. Two misses are unexplained: Cavour’s power-off curve rises faster than hpr’s through subsonic flow, from −8.3% at Mach 0.3 to −12.6% at 0.85; and Juno III’s stays flat up to Mach 0.91, where hpr’s has begun its rise, its nose’s and its boattail’s, +19.3% at Mach 0.85 and +32.0% at 0.9.

What the boattail’s wave drag changed. Calisto ends in a short, steep conical boattail: 0.47 calibres long, narrowing to 69% of the diameter, a slope of 18.4°. Until M1.8b3 hpr gave it only a share of the base drag, 0.083 at Mach 1.2 and 0.050 at 2.0, and read −29.8% to −24.4% from Mach 1.2. Its supersonic wave drag (Boattails faster than sound) is 0.283 at Mach 1.2, 0.190 at 1.5 and 0.120 at 2.0, and its base drag falls by a third; together they close the gap from 0.204 to 0.035 at Mach 1.2 and from 0.128 to 0.077 at 2.0. RASAero II lists such drag as its own term, “other body wave” drag. Calisto’s 18.4° boattail is steeper than any attached boattail measured here, and 16° boattails read 26% to 54% high, so this agreement is not support for the model at that angle. What is left grows with Mach number, and part of it is hpr’s body, which reads 6% to 10% low faster than sound against a worked example with every input known (below).

The unrecorded inputs now span most of the rest. Calisto’s fins could be square, rounded or an airfoil, 2 to 6.35 mm thick, smooth or painted (tests::calistos_rows_by_fin_and_finish). The committed inputs, square, 3 mm and smooth by the rule of the Mach 0.3 check, have 15, 3 and 8 rows within 10% by band. Rounded fins 4.76 mm thick, smooth, have 15, 4 and 14; airfoil fins 6.35 mm thick, smooth, have 11, 4 and 17. No combination has every row within 10%. Before the wave drag no combination had rows within 10% both below Mach 0.8 and from Mach 1.2. So most of what is left is within what the unrecorded inputs span; hpr keeps the stated rule rather than picking the inputs that fit.

Drag against MIL-HDBK-762’s sample calculation

RASAero II’s curves can’t show which way hpr leans, because their inputs are guessed. A reference with every input known can. MIL-HDBK-762, the U.S. Army’s handbook for designing unguided rockets, works one rocket’s drag through by its own methods, term by term, from Mach 0.5 to 3.2 ([762] Table 5-4, pp. 5-58 to 5-66). The rocket is 3.84 m long and 0.16 m across. It has a 3-calibre tangent ogive nose, a plain cylinder with no boattail, and four fins 0.32 m long, 51 mm tall and 6.4 mm thick, flush with the base (Fig. 5-155).

This is a calculation, not a measurement: it checks hpr’s methods against another set of methods, whose base drag comes from measured bases. The table is transcribed with its pages in mil-hdbk-762-sample-drag.json, and every row sums to its printed total. The rocket is validation/designs/mil-hdbk-762-sample-rocket.json, with a smooth finish, as the handbook’s friction is. hpr flies it at the table’s Reynolds numbers. tests::drag_against_mil_hdbk_762_sample recomputes the comparison in drag-vs-mach.json and pins the rows within 10%. The 10% target comes from M1.8. hpr’s numbers for this rocket were seen before it was chosen as a reference, so this is not a blind test.

The fins are left out. The handbook draws each fin as a single wedge, sharp at the leading edge and blunt at the trailing edge, and gives the fins a thin wedge’s wave drag and the base drag of their trailing edges. hpr has no such section. The design gives them square edges, whose leading edges hpr charges the pressure of air brought to a stop against them (0.100 at Mach 2, against the handbook’s 0.016 for the whole fin). So each side’s fin pressure drag is shown but left out of the totals compared; the fins’ friction stays in.

Each term, the handbook’s first and then hpr’s, on the reference area:

Six of twelve rows are within 10%. hpr reads +12.3% to +31.9% high from Mach 0.9 to 1.2, and −6.0% to −9.6% low from Mach 1.6:

• The nose through Mach 1. Niskanen’s ogive gives two to three times the handbook’s: 0.164 against 0.052 at Mach 1.0, and 0.234 against 0.109 at 1.1. Stoney’s measured 3:1 cone also sits under Niskanen’s closed form through the rise (Drag through Mach 1). From Mach 2 the two agree within 10%.
• The base. Niskanen’s base drag (eq. 3.94, after Fleeman’s missile design textbook) gives 0.250 at Mach 1.0 where the handbook reads 0.183 from measured bases. Faster than sound hpr’s is the lower: 0.125 against 0.147 at Mach 2.
• Friction reads 10.4% to 15.9% lower in hpr. The handbook takes a smooth flat plate’s friction and adds 15% on the body; hpr’s body factor ([N09] eq. 3.85) adds 2% for this slender body, which accounts for about 11 points. The rest is unexplained; the two methods correct friction for Mach number differently.

So hpr’s body reads high through Mach 1, from the nose and the base, and 6% to 10% low faster than sound, from friction and the base. That is the same sign as Calisto’s gap to RASAero II, a third of its size. This rocket has no boattail, so it says nothing about a boattail’s own drag.

Roll against the Arcas Robin and the Basic Finner

What is checked: hpr’s roll forcing against NASA’s measured roll effectiveness of the two Arcas Robin models from Mach 1.5 to 4.63 (TN D-4014 Fig. 14, [D4014]), and its roll damping against the Basic Finner’s measured from Mach 1.5 to 3.0 and Barrowman’s own computed value at Mach 0.07 ([B67] Figs. 5-6 and 5-7). The readings are in the wind-tunnel fixture and the Basic Finner’s; cargo xtask aero writes the comparison to the roll fixture, and hpr_aero::tests::roll_against_mach recomputes every row. The roadmap set no target.

The forcing. C_lδ per degree of cant, on the body’s cross-section and diameter, at an angle of attack of 0 (the reports’ symbol is per degree; the models’ fins were canted 2°):

The two models differ only in the body’s length ahead of the fins, which hpr’s forcing doesn’t see; the measured values differ by up to 0.006 per degree (8%, at Mach 4.63), and the report calls the effectiveness “about the same for either vehicle” ([D4014] p. 6). The short model’s readings were corrected when roll was added: the first reading had put each of its panels’ zeros 0.005 to 0.007 above the grid line it lies on (ADR-031). From Mach 2.3 all 8 are within 5.3%. At Mach 1.5 and 1.8 hpr reads high, as Barrowman found for another sounding rocket: linear theory’s load climbs toward Mach 1 faster than the fins’ does. Without the body factor k_T(B) (0.935 here) every value would be 7% higher.

The damping. C_lp of the Basic Finner, four square fins one diameter in chord and span on a body one diameter across, per unit of p d/(2V):

hpr reads low faster than sound, more so as the Mach number grows. Barrowman’s own curve, from Busemann’s third-order expansion ([B67] eq. 3-7, a higher-order theory that counts the fins’ thickness), is 5.68% from the same points on average ([B67] p. 66); first-order theory, hpr’s, reads low partly for want of a term for the fins’ 8% thickness. At Mach 0.07 hpr gives his computed value within 2.0%, which is how hpr’s reading of his damping method, the fin’s own slope over the strips, was checked (above).

The flight (hpr_sim::tests::canted_fins_spin_to_the_analytic_balance): Valetudo with 1° of cant at 100 m/s, with no drag and no gravity, settles on the closed-form steady roll rate of the worked example, −16.948 rad/s, within 1e-6 (the test’s bound; 1e-11 measured), and one time constant in is within 1e-5 of the exponential approach (2e-10 measured); no pitch or yaw appears.

The pieces (hpr_aero::fins::tests): the polygon’s span moments against the trapezoid’s and the ellipse’s integrals ([N09] eq. 3.70–3.71); the supersonic forcing and damping against a 20,000-strip sum of the same load on the Arcas Robin’s swept fin, within 1e-7, from Mach 1.5 to 4.63; the subsonic ones against Barrowman’s closed forms, and both continuous at Mach 0.8 and M_s; k_R(B) against its integral (eq. 3-121) by Simpson’s rule within 1e-10.